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THEORETICAL PHYSICS
THEORETICAL PHYSICS
BY
W. WILSON, F.R.S.
Vol. I. Mechanics and Heat. Newton — Carnot.
Vol. II. Electromagnetism and Optics. Maxwell — Lo rentz .
Vol. Ill, Relativity and Quantum Dynamics. Einstein — Planck.
THEORETICAL PHYSICS
BY
W. WILSON, F.R.S.
HILDRED CARLILE PROFESSOR OF PHYSICS IN THE UNIVERSITY OF LONDON, BEDFORD COLLEGE
VOL. I
MECHANICS AND HEAT
NEWTON— CARNOT
WITH EIGHTY DIAGRAMS
NEW YORK
E. P. BUTTON AND COMPANY INC.
PUBLISHERS
■c5^3t)
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PRINTED IN GREAT BRITAIN
^
PREFACE
THE purpose of the present work is to present an account of the theoretical side of physics which, without being too elaborate and voluminous, will nevertheless be sufficiently comprehensive to be useful to teachers and students. This, the first volume, deals with mechanics and heat ; the second volume will be devoted to electromagnetism and optics and possibly the introductory part of relativity ; while the remaining volume will deal with relativity and quantum dynamics. The contents of this part are based on the notes of lectures delivered at one time or another at Bedford College and King's College (London). In selecting the subject-matter I have been influenced chiefly by its importance from the point of view of exhibiting the unity of physical theory and in a secondary degree C" by any special interest, historical or other, commending it, or ^|V^by its suitability as a means of preparing the ground for more important things to follow. The unavoidable incompleteness is compensated to some extent by the bibliographical references T' and notes appended to many of the chapters. ^ Each part of the subject is developed in a way which follows,
f. broadly speaking, its historical growth, and this first volume is fCs entirely ' classical ', the dynamical part of it being based on the foundations of Newton. At the same time every opportunity • that presents itself is utilized to open the way for the description T of modern developments of physical theory which will occupy -~ later parts of the work. The methods of elementary vector and "C tensor calculus are introduced at the outset and consistently V followed, partly on account of their fitness and utility, and J^ partly as an introduction to a more complete account of tensor
calculus which will have a place in a later volume. *^ Care has been taken to make the nature of fundamental ^ principles as clear as possible and everything is developed from ^ the simplest beginnings. No very serious demands are made • on the mathematical equipment of the reader. A certain ac- ^^ quaintance with the elements of the calculus and analytical
y^ V
vi THEORETICAL PHYSICS
geometry is assumed and any mathematical methods which extend beyond this are explained as they may be required.
Needless to say, I have derived much assistance from many classical works and papers ; in fact from most of those mentioned in the bibhographical appendices. I am indebted to Dr. Maud 0. Saltmarsh for reading the proofs.
W. W.
January i 1931
CONTENTS
PREFACE
GENERAL INTRODUCTION
PAGE V
CHAPTER I FOUNDATIONS OF EUCLIDEAN TENSOR ANALYSIS
SCALAHS, VeCTOBS Aiq^D TeNSORS
Scalar and Vector Products . Co-ordinate Transformations . Tensors of Higher Rank Vector and Tensor Fields
4 4 5
9 11 12
CHAPTER II
THE THEOREMS OF GAUSS, GREEN AND STOKES. FOU- RIER'S EXPANSION 16
Theorem of Gauss ........ 16
Green's Theorem . . . . . . . .18
Extensions of the Theorems of Gauss and Green . 22
Theorem of Stokes ........ 23
Fourier's Expansion . . . . . . .28
Examples of Fourier Expansions ..... 32
Orthogonal Functions . . . . . . .36
CHAPTER III INTRODUCTION TO DYNAMICS 39
Force, Mass, Newton's Laws ...... 39
Work and Energy ........ 42
Centre of Mass ........ 44
Path of Projectile ........ 45
Motion of a Particle under the Influence of a Central
Attracting or Repelling Force . . . .47
Angular Momentum of a System of Particles free from
External Forces ....... 48
Planetary Motion ........ 50
Generalized Co-ordinates . . . . . .57
Moments and Products of Inertia ..... 60
The Momental Tensor ....... 63
Kinetic Energy of a Rigid Body . . . . .63
The PENDULuiki . . . . . . . . .65
vii
VIU
THEORETICAL PHYSICS
CHAPTER IV DYNAMICS OF A RIGID BODY FIXED AT ONE POINT
Euler's Dynamical Equations . Geometrical Exposition . Euler's Angular Co-ordinates The Top and Gyroscope . The Precession of the Equinoxes
page
72 72 78 82 84 91
CHAPTER V
PRINCIPLES OF DYNAMICS 92
Principle of Virtual Displacements .... 92
Principle of d'Alembert ....... 95
Generalized Co-ordinates ...... 97
Principle of Energy . . . . . . .100
Equations of Hamilton and Lagrange . . . .102
Illustrations. Cyclic Co-ordinates .... 105
Principles of Action . . . . . . .109
Jacobi's Theorem . . . . . . . .116
CHAPTER VI WAVE PROPAGATION ....
Waves with Unvarying Amplitude . Waves with Varying Amplitude Plane and Spherical Waves . Phase Velocity and Group Velocity Dynamics and Geometrical Optics .
121 121 131 132 136 138
CHAPTER VII ELASTICITY 141
Homogeneous Strain ....... 141
Analysis of Strains ....... 146
Stress .......... 153
Stress Quadric. Analysis of Stresses . . . .156
Force and Stress . . . . . . . .159
Hooke's Law. Moduli of Elasticity . . . .162
Thermal Conditions. Elastic Moduli of Liquids and
Gases 166
Differential Equation of Strain. Waves in Elastic
Media 168
Radial Strain in a Sphere . . . . . .171
Energy in a Strained Medium . . . . .174
Equation of Continuity. Prevision of Relativity . 175
CHAPTER VIII HYDRODYNAMICS ....
Equations of Euler and Lagrange Rotational and Irrotational Motion Theorem of Bernoulli The Velocity Potential . JCdstetio Energy in a Fluid
180 180 185 186 189 192
CONTENTS
IX
Motion of a Sphere through an Incompressible Fluid Waves in Deep Water ......
Vortex Motion .......
page 192 196 198
CHAPTER IX MOTION IN VISCOUS FLUIDS
Equations of Motion in a Viscous Fluid Poiseutlle's Formula .... Motion of a Sphere through a Viscous Liquid OF Stokes ......
Formula
203 203 205
209
CHAPTER X
KINETIC THEORY OF GASES .217
Foundations of the Kinetic Theory. Historical Note 217
Boyle's Law 218
Laws of Charles and Avogadro. Equipartition of Energy 221
Maxwell's Law of Distribution ..... 225
Molecular Collisions. Mean Free Path . . . 232
Viscosity. Thermal Conductivity ..... 236
Diffusion of Gases ........ 240
Theory of van der Waals ...... 244
Loschmidt's Number ....... 253
Brownian Movement . . . . . . .255
Osmotic Pressure of Suspended Particles . . . 256
CHAPTER XI STATISTICAL MECHANICS 259
Phase Space and Extension in Phase . . . .259
Canonical Distributions ....... 262
Statistical Equilibrium of Mutually Interacting Systems 263 Criteria of Maxima and Minima . . . . .267
Significance of the Modulus . . . . .267
Entropy .......... 270
The Theorem of Equipartition of Energy . . . 270
CHAPTER XII
THERMODYNAMICS. FIRST LAW 272
Origin of THERMODYNA]\ncs ...... 272
Temperature ......... 272
Equations of State ........ 275
Thermodynamic Diagrams ...... 276
Work Done During Reversible Expansion . . . 278
Heat 280
First Law of Thermodynamics . . . . .280
Internal Energy of a Gas . . . . . .282
Specific Heat ......... 282
The Perfect Gas 283
Heat Supplied to a Gas During Reversible Expansion . 286
THEORETICAL PHYSICS
PAGE
CHAPTER XIII
SECOND LAW OF THERMODYNAMICS . . . . 288
The Perpetuum Mobile op the Second Kind . . . 288
Cabnot's Cycle ........ 289
Carnot's Principle ........ 290
Kelvin's Work Scale of Temperature . . . .292
The Work Scale and the Gas Scale . . . .296
Entropy .......... 297
Entropy and the Second Law op Thermodynamics . 298
Properties of the Entropy Function. Thermodynamics
AND Statistical Mechanics ..... 300
CHAPTER XIV
THE APPLICATION OF THERMODYNAMICAL PRINCIPLES 304
General Formulae for Homogeneous Systems . . 304
Application to a v. d. Waals Body ..... 308
Thermodynamic Potentials ...... 309
Maxwell's Thermodynamic Relations . . . .311
The Experiments of Joule and Kelvin and the Realization
OF THE Work Scale of Temperature . . . 312
Heterogeneous Systems . . . . . . .317
The Triple Point ........ 317
Latent Heat Equations . . . . . . .318
The Phase Rule . . . . . . . .320
Dilute Solutions ........ 323
INDEX OF SUBJECTS 327
INDEX OF NAMES 331
THEORETICAL PHYSICS
§1. GENERAL INTRODUCTION
PHYSICAL science, in the restricted sense of the term, is concerned with those aspects of natural phenomena that are regarded as fundamental. Broadly speaking it inves- tigates things with which we are brought into immediate contact through the senses, hearing, touch and sight ; and it includes, among others, the familiar sub-divisions sound, heat and light. But sense perceptions themselves hardly enter into physics. Indeed, they are deliberately excluded, as far as may be, from physical investigations. Spectra are observed photographically and the colour of a spectral line is really not a thing in which the physicist takes any interest. No great effort of imagination is needed to conceive the possibility of photometric devices whereby a completely blind observer might carry out for himself aU the observations on a spectrum which have any significance for physics. Temperature is not measured by feeling how warm a thing is, nor in acoustical investigations do we rely on the sense of hearing. In fact, the use of human senses is practically con- fined, in experimental physics, to the observation of coincidences, such, for example, as that of the top of the mercury column in a thermometer with a mark on the scale of the instrument, or that of the spider line in a telescope with a star or a spectral line, and the associated coincidence which gives the scale reading.
Physical science is the cumulative result of a variety of closely correlated activities which have given us, and are adding to, our knowledge of what we shall call the Physical World, and the present treatise is an attempt to present, in outline, a con- nected account of the body of doctrine which has grown out of them.
There are three well-defined periods in the development of the theoretical side of physics since the time of Galileo. The earliest of these, which we may call the ' matter and motion ' period, came to an end in 1864 when Clerk Maxwell's electro-
1
2 THEORETICAL PHYSICS
magnetic theory of light appeared.^ The physicist of this period conceived the world as built up, roughly speaking, of minute particles (atoms) endowed with mass or inertia and capable of exerting forces (gravitational, electric, etc.) on one another. Their behaviour and mutual interactions were subject to certain djmamical principles, summarized in Newton's laws of motion. A phenomenon was considered to be satisfactorily accounted for when it could be represented as a mechanical process ; when it could, as it were, be reproduced by mechanical models differing merely in scale from something that might be constructed in a workshop. This mechanical physics was extraordinarily successful, and was tenaciously adhered to and defended, even so recently as the opening years of the present century, as the following quotation from the preface to the first edition of an admirable work on the theory of optics ^ will show.
' Those who believe in the possibiHty of a mechanical conception of the universe and are not wiUing to abandon the methods which from the time of GaHleo and Newton have uniformly and exclusively led to success, must look with the gravest concern on a growing school of scientific thought which rests content with equations correctly represent- ing numerical relationships between different phenomena, even though no precise meaning can be attached to the symbols used.'
The second period, from 1865 till the opening years of this century, has a transitional character. Maxwell's theory (which, it may be remarked, united the previously disconnected provinces of light and electricity) led eventually to the abandonment of the effort to establish electrical phenomena on the old-fashioned ' matter and motion ' basis and placed ' electricity ' on equal terms by the side of ' matter ' as a building material for the physical world. In the 'eighties indeed the most characteristic property of matter, namely mass or inertia, was successfully accounted for in electrical terms, and attempts began to be made (with some success) to provide a purely electrical basis for theo- retical physics.
Distinguishing marks of the present period of theoretical physics (since 1900) are the development of the quantum and relativity theories and the consequent overthrow of the sover- eignty of Euclidean geometry and Newtonian dynamics. These latter, however, retain their practical importance in almost undiminished measure, and it would indeed be inaccurate to
^ It is noteworthy that Maxwell wrote a little book called Matter and Motion which, though he was the inaugurator of a new epoch in physics, presents a very fair picture of, and indicates his sympathy with, the ideals and aims of the earlier period.
^ Schuster : An Introduction to the Theory of Optics (Arnold, 1904).
GENERAL INTRODUCTION 3
speak of them as untrue or disproved ; but they now appear as limiting cases of the more comprehensive modem theories. The old problems of the explanation of electrical phenomena in mechanical terms, or of matter in electrical terms, have no longer any significance, and physical theory is approximating more and more to a vast and unified geometrical structure such as was not dreamt of in the philosophy of Euclid or Newton.
CHAPTER I FOUNDATIONS OF EUCLIDEAN TENSOR ANALYSIS
§ 2. SCALABS, VeCTOES AND TeNSOES
A PHYSICAL quantity which can be completely specified by a single numerical statement (the unit of measure- ment having once been chosen) is called a scalar. Examples of scalars are electric charge, mass, temperature, energy and so forth. A vector is a physical quantity associ- ated with a direction in space. For its complete specification three independent numerical statements are necessary. The typical example of a vector is a displacement. If a small
body or particle is given a series of ° displacements represented by AB, BC,
^ CD, DE (Fig. 2), which are not neces- sarily co-planar, it is obvious that these bring about a result equivalent to the single displacement represented Fig. 2 by AE. The displacement AE is
called the resultant of the dis- placements AB, BC, CD, DE. Any vector can be represented in magnitude and direction by a displacement. Examples of vectors are force, velocity, momentum, electric field intensity and so on. If, for instance, four forces are applied to a body (to avoid irrelevant complications we shall suppose them all to be applied at the same point in the body) and if a straight line AB (Fig. 2) be constructed having the direction of the first force and a length numerically equal to it in terms of some convenient unit, and if a second straight line BC be drawn to represent the second force in a similar way and so on ; then the four forces are equivalent to a single or resultant force which is represented in magnitude and direction by AE. The three independent numerical data which are necessary to express the vector completely may be given in various ways. We may, for example, give the absolute value of the vector, i.e. the length of the line AB or BC (Fig. 2), representing it ; in which case we have to give two additional numerical data to fix its direction
§21] EUCLIDEAN TENSOR ANALYSIS 5
relative to whatever frame of reference we may have chosen. The three independent data, however they may be chosen, are called the components of the vector. It is usual, however, to restrict the use of the term ' component ' in the way indicated in the following statement : Any vector can be represented as the resultant of three vectors which are parallel respectively to the X, Y and Z axes of a system of rectangular co-ordin- ates. These three vectors are called its components in the X, Y and Z directions. Unless the contrary is stated, or implied by the context, we shall use the term ' component ' in this more restricted sense. If A represents the absolute value of a vector, we shall represent its components by A^, Ay and A^ and refer to it as the vector A, or the vector (A^, Ay, A^). The statement ' A^, Ay and A^ are the X, Y and Z components of the vector A ' may conveniently be expressed in the abbreviated form :
A ^ (Ag., Ay, A^).
It is clear that when a vector is represented by a line drawn from the origin, 0, of a system of rectangular co-ordinates to some point P, its components are the co-ordinates of P.
Besides scalars and vectors we have still more complicated quantities, or sets of quantities, called tensors. A tensor of the second rank requires for its complete specification 9 or 3^ inde- pendent numerical data, which are not necessarily aU different. Just as a vector can be represented by a displacement, so can a tensor of the second rank be represented, in its essential pro- perties, by a pair of displacements. This will be more fully explained later. The state of stress in an elastic solid is an example of such a tensor. It has become customary in recent times to use the term ' tensor ' for all these different types of physical quantities. A scalar is a tensor of zero rank ; it requires for its specification 3° or 1 numerical datum. A vector is a tensor of the first rank, requiring for its specification 3^ independent numerical data and so on.
§ 2-1. Scalar and Vector Products
The inner or scalar product of two vectors is defined to be a scalar quantity numerically equal to the product of their absolute values and the cosine of the angle between their direc- tions. If the absolute values are A and B, and if the included angle is d, the scalar product is AB cos 6. It is convenient to abbreviate this expression by writing it in the form (AB) or (BA). A very important instance of a scalar product is the work done by a force when its point of application is displaced.
6 THEORETICAL PHYSICS [Ch. I
If, for example, A represents a force (which we shall suppose to be constant), B the displacement of the point where it is applied, and 6 the angle between their directions, the work done is expressed by AB cos 6 or briefly by (AB).
To elucidate the properties of the scalar product it is con- venient to represent the vectors A and B by displacements from
the origin O of rectangular co- ordinates (Fig. 2-1). Let the ter- minal points, p and q, of the dis- placements be joined by a straight line, the length of which is repre- sented by t. Then we have
t^ = A^ + B^ - 2AB cos 6.
Since the co-ordinates of p and q are (A^, Ay, A J and (B^, B^, B^) respec- tively, it is evident that t is the diag- onal of a parallelopiped, the edges of which are parallel to the axes X, Y and Z and equal respectively to \A^ — BJ^, \Ay — By\ and \A^ — Bj, the symbol \x \ being used to represent the absolute value of x. Therefore
P = {A, - B,)^ + (A, - B^y + (A, - B,r. If we remember that
A^ = AJ-{-A/ + A,^ .... (2-10) and B^=BJ + B/ + B,^
we find on equating the two expressions for t^
Fig. 2-1
ABcos0=:^A+^A+^A . . (2-11)
This important result may be expressed in words as follows; — If the like components of two vectors are multiplied together, the sum of the three products thus formed is equal to the scalar product of the two vectors.
When the angle between the two vectors is a right angle it is obvious that
^A + ^A + ^A = 0 . . ._ . (2-12)
and conversely, when equation (2*12) holds the directions of the two vectors must be at right angles (if we except the trivial case where one or both of the vectors are equal to zero). If we refer the vectors A and B to new rectangular co-ordinates, in which their components are
a;, A^', a:, and BJ, B/, B/, the scalar product will now be
ajb,'+a;bj+a:b:
§21] EUCLIDEAN TENSOR ANALYSIS 7
and we must have
A A + AyB, + A A = AjBj + a;b; + a:b:,
since the value of the scalar product is clearly independent of the choice of co-ordinates. We have here an example of an invariant, i.e. of a quantity which has the same numerical value whatever system of co-ordinates it may be referred to.
The product of the absolute values of the two vectors and the sine of the angle included between their directions is called their outer or vector product. In the case of the vectors A and B (Fig. 2-1) we have
vector product = AB sin 9
We shaU usually abbreviate this expression by writing it in the form [AB].
Squaring both sides of (2-11) we have
A2B2 - A2B2 sin^ d = {A,B, -F A,B^ -\- A,B,Y or, by (2-10) A^B^ sin2 d = (^,2 _j_ A/ + ^/)(5,2 + 5/ + 5^2)
-(^A + ^A+^A)^ On multiplying out, we easily recognize that this last equation is equivalent to
A^B^ sin^ e = {A,B, - Afi,)^ + (Afi, - A,B,)^
+ (A^,-A,B,)^ . . . .(2-13)
Obviously we may change the sign in any of the expressions AyB^ — Afiy, etc. on the right without affecting the equation. This ambiguity is intimately associated with a corresponding feature in rectangular axes of co-ordinates, and it now becomes necessary to give a precise specification of the type of rectangular axes we propose to use. We shall do this in the following terms : The motion of an ordinary or right-handed screw travelling along the X direction turns the Y axis towards the Z axis. In this description the letters X, Y and Z may, of course, be interchanged in a cyclic fashion. It is evident from equation (2*13) that the three quantities,
^x = ^y^z — ^ A' ^y = ^ A — ^x^z5 ^z = ^ A — ^ A
can be regarded as the components of a vector the absolute value of which is AB sin d. The question arises : What is the relation between the directions of the vectors a, A and B ? The scalar product (a A)
= a^^ -I- GyAy -\- a,A,
= (AyB, - A,By)A, + (^A - ABMy + (^ A - A^x)A = 0 identically. 2
8
THEORETICAL PHYSICS
[Ch. I
It follows that the vector o is at right-angles to A and by forming the scalar product (oB) we can show further that o is also at right angles to B. Let us turn the co-ordinate axes about the origin so that the vectors A and B lie in the XY plane in the way
indicated in Fig. 2-11. The components a^ and ay will now be zero, and we see that a^ is positive, since A^By — AyB^ is obviously greater than zero.
This means that when the co-ordinate axes are placed in this way relatively to the vec- tors A and B , the vector o will be in the direction of the Z axis, and so we conclude that the motion of an ordinary or right-handed screw travel - Hng in the direction of o turns the vector A towards the vector B. We shall extend the use of the notation [AB] to represent the vector product completely, i.e. both in magni- tude and direction. That is to say [AB] means the vector, the X, Y and Z components of which are respectively
AyB, - Afiy, A,B, - A,B„ A^y - AyB,,
and [BA] means the vector
(ByA, - BAy, BA. - BJL,, B^y - ByA,), which has the opposite direction.
The scalar product of any vector G and [AB] is
(G[AB]) = ClAB], + Oy[AB]y + ClABl, (G[AB]) =. GMyB. - A,By) + Cy(A,B, - A^ *
+ CM^y - AyB,),
Fig. 211
or (G[AB])
A.. A.
Bx, By, B
(2-14)
Clearly this determinant is an invariant, since a scalar product is an invariant. If again we imagine the axes of co-ordinates to be turned about the origin till the vectors A and B lie in the XY plane, as in Fig. (2-11), the scalar product (G[AB]) becomes C^[AB]^, since the X and Y components of [AB] are both zero. Therefore
(G[AB]) = G cos £ AB sin (9 . . . (2-15)
where s is the angle between the directions of G and of the Z axis. If therefore e is less than -, the scalar product (G[AB])
§2-2]
EUCLIDEAN TENSOR ANALYSIS
9
or the determinant (2*14) is equal to the volume of the parallelo- piped which is determined by the displacements A, B, G.
We may formulate this result as follows : If the motion of an ordinary or right-handed screw travelling along the direction of C turn A towards B, then the determinant (2* 14) is equal to the volume of the parallelepiped deter- mined by the vectors A, B and G.
Obviously we may interchange A, B and G in cyclic fashion in this theorem.
If cos £ in (2*15) is zero, the vectors G and [AB] are at right- angles to one another ; but this means that A, B and G are in the same plane and on the other hand that the determinant (2*14) is zero. In fact, if A, B and G are all different from zero, the necessary and sufficient condition that they shall be co-planar is :
= 0
(2-16
§ 2-2. CO-ORDIKATE TRANSFORMATIONS
Let X, Y, Z and X', Y', Z' be two sets of rectangular axes of co-ordinates with a common origin 0 ; and let P be any point, the co-ordinates of which are x, y, z and x' , y\ z' in the two systems respectively (Fig. 2-2). Let us further represent the cosines of the angles be- tween X' and X, Y, Z, by Z^,, \ and l^ respectively ; those be- tween Y' and X, Y, Z by m^, tYiy and m^ respectively, and so on. The problem before us is : given ic, y^ z, the co-ordinates of P in the system X, Y, Z, to find x\y\z\ its co-ordinates in the other system X', Y', Z', and vice versa. Drop a perpendicular Pm on OX', so that Om is equal to x' , the X' co-ordinate of P in the system X', Y', Z'. We may regard both OP and Om as vectors and we have clearly
?^;jJ;^;of)systemX',Y',Z'.
The rule (2*11) gives us for their scalar product the alternative expressions,
10 THEORETICAL PHYSICS [Ch. I
x\x + x'l^y + x'l^ (System X, Y, Z)
and x'^ (System X', Y', Z').
On equating these two expressions, and dividing by the common factor x', we finally obtain
x' = l^x + l^y + %z.
In a similar way we may show that
y' = m^x + m^y + m,z,
z' = n^x -j-riyy '\-nz^. - - - - \ )
The equations of the inverse transformation are easily found to be
X = l^x' + m^y' + n^z'
y = l^x' +m^y' + n^z' .... (2-21)
z = l^x' + my + n^z\
We may, evidently, regard l^,ly and l^ as the components of a unit vector (i.e. a vector the absolute value of which is unity) in the system X, Y, Z. A similar remark applies to (m^, m^, mj and {n^, Uy, n^). And in the system X', Y', Z' we may regard (h^ ^x. ^x). ik^ ^y' ^1/) aiid (?„ m„ rij as unit vectors.
For many purposes it is convenient to represent these direc- tion cosines by a single letter, distinguishing one from another by numerical subscripts, thus :
(^x5 ^y ^z) ^ (^iij ^12) Ctiajj
(n^, Uy, n,) = (a.
All six equations of transformation given above are con- veniently represented in the following schematic form : —
. (2-22)
Mathematically a vector may be defined as a set of three quantities v^hich transform according to the rules em- bodied in (2-22).
There are certain important and interesting relations between the direction cosines a. For example, the sum of the squares of the a's in any horizontal row, or in any vertical column of (2*22) is equal to unity :
|
X |
y |
z |
|
|
x' |
ail |
ai2 |
ai3 |
|
y' |
^21 |
"22 |
"23 |
|
z' |
ttsi |
"32 |
^33 |
etc.
ail + aia^ _|_ a^32 ^ j^ ai2^ + a22^ + asa^ = 1
. (2-23)
§2-3] EUCLIDEAN TENSOR ANALYSIS 11
The correctness of these equations is obvious, since in each case the left-hand member can be regarded as the sum of the squares of the components of a unit vector. Further, the sum of the products of corresponding a's in any two horizontal rows, or in any two vertical columns is zero, e.g.,
aiittis + OLiiCn^z + a3i«33 = 0,
a2ia3i + a22a32 + a23a33 = 0 ... (2-24) and so on. These equations follow since the left-hand member in each case can be regarded as the scalar product of two unit vectors which are at right angles to one another. Finally we have the relation
= 1 (2-25;
Ctll, «12j 0^13 0^21? Ct225 Ct23 «31j «S2j Ct33
since by (2*14) this determinant represents the volume of the paraUelopiped bounded by the three mutually perpendicular unit vectors (Z^, ly, \); (m^, m^, mj and (n^, riy, n,),
§ 2-3. Tensors of Higher Rank
We are now able to define more precisely a tensor of higher rank than a vector. Take, for example, a tensor of the second rank, such as that which expresses the state of stress in an elastic solid. It is a set of 3^ quantities, called its components,
Pxx^ Vxy^ Vxzy jPyxi jPyyi Pyz) Pzx^ Pzyj Pzz^
having the property that the values of the components p^', p^y, etc., in the system X'Y'Z' are calculated from p^^, p^y, etc., the components in the system X, Y, Z, by precisely the same rules as those for calculating AJBJ, AJBy, etc., from the pro- ducts AJB^, AJBy, etc., where A^, Ay, By, etc., are the components of two vectors. A tensor of the second rank is said to be sym- metrical when the subscripts of a component may be interchanged, e.g., when
Pxy JPyx
The system of stresses in an elastic solid in equilibrium consti- tutes such a symmetrical tensor. If, on the other hand,
jPxy Pyx^
the tensor is said to be anti-symmetrical. Since in this case,
Pxx Paxa^
the components p^, Pyy, etc., with two like subscripts wiU all three be zero. As an example of an anti-symmetrical tensor we
12 THEORETICAL PHYSICS [Ch. I
may instance that formed from two vectors A and B in the following way : —
|
AA |
-AA, |
AA |
- A A |
A A |
-AA |
|
AB. |
-AJi,, |
AyB, |
- A A. |
A A |
- ^ A. |
|
AB. |
-AA |
AA |
- A A. |
AA |
- ^A- |
Its XX, YY and ZZ components are zero and the remaining six are the components of the vectors [AB] and [BA]. In fact we may dispense with these vector products by employing this tensor.
More generally, if we have n vectors and select one component of each and multiply them together, the 3^* products obtained from all the possible selections constitute a typical tensor of the nth rank, and any set of 3'^ quantities will constitute a tensor of rank n if they obey the same laws of transformation as the 3'* components of the typical tensor.
§ 24. Vector and Tensor Fields
We shall often be concerned with regions in which electric, magnetic or gravitational forces manifest themselves. We call such regions fields of force. They are characterized in each of these examples by a vector which varies continuously from point to point in the region and which may be termed the intensity of the field. In hydrodynamics we are concerned with regions filled with a fluid, the motion of which can be described by giving its velocity at every point in the region. In all these examples we may use the general term vector field for the region in question. Or we may be concerned (e.g. when we are studying the state of stress in an elastic solid, or the Maxwell stresses in an electrostatic field) with the components of a tensor of higher rank than a vector and with the way in which they vary from one point in the field to another. In such a case we may call the region a tensor field.
The description and investigation of vector or tensor fields involves the use of partial differential equations, and we shall
O O O
therefore study some of the features of the operations -^^ ■^, -^,
ox cy cz
where the round cZ's are the conventional symbols for partial
differentiation, i.e. -^r- means a differentiation in which the other
ox
independent variables y, z and the time are kept constant. In
O o o
the first place we may show that -^^, ^r-, ;:-- have the same trans- Bo; 31/ dz
formation properties as the X, Y, Z components of a vector.
§2-4] EUCLIDEAN TENSOR ANALYSIS 13
Let ^ be any quantity which varies continuously from point to point. Then by a well-known theorem of the differential calculus,
dx' dx dx' dy dx' dz dx'
But by (2-22)
X = a^^x' + aay + o.^^z\
y = aiao;' + aaa^/' + ctsi^',
z = ttiso;' + a^sy' + cLsz^', therefore
dx _ ^y _ ^^ _
and on substituting in (2 •4) we have
dcf) _ ^</> I ^^ \ ^^
cx ox dy oz
or, dropping </>, we have the equivalence,
^-, = ttii^ + ai2^ + ai3-- . . . (2-41) ox ox oy oz
This is sufficient to establish the vectorial character of these operations.
If A = {Ay., Ay, A^) is a field vector, the quantity
dA^ dAy dA, dx dy dz
wiU be an invariant since it has the same transformation pro- perties as a scalar product. It is called the divergence of the vector A and is written
div A.
Furthermore the three quantities
dA^ dAy dA^^ dA^ dAy dA^
dy dz ' dz dx ' dx dy
must be the X, Y and Z components of a vector, since they have the same transformation properties as the components of a vector product. This vector is called the curl of A or the rotation of A and is written
curl A or rot A.
It is easy to show that
div curl A = 0, . . . . . (2-42) where A is any field vector. We have in fact,
r) 7) r)
div curl A = — {curl A}^ + {curl A}^ -|- ^{curl A}^ ox cy oz
14 THEORETICAL PHYSICS [Ch. I
d* 1 A = —f—' - —A -4- —f^-^ - ^^A
dx\ dy dz ) dy\ dz dx )
'bzx dx dy
d (dA^ _ dA^\
It will be seen that this is identically zero.
If A = (^^, Ay, Ag) is a vector and ^ any scalar quantity, it is obvious that
iA^, ^,0, A,ct>) is a vector. Similarly
/dcj) dcf) 8</>\ \dx' dy' dz) is a vector. Such a vector is called the gradient of the scalar quantity and is written grad ^. We have therefore
^^^'^-(^i'%t)- ■ ■ ■ t^-^'
The components of the vector curl grad ^ are all identically zero. Take the X component for example :
{curl grad ^ L = ;g- (grad */» L - ^ (grad ^ }y
d {dci>) d(dci>'
{curl grad ^L = ^^^^^^ a.^a^
= 0 identically. We may write this result in the form
curl grad ^ = 0 . . . . . (2-431)
rl r) f)
The quasi- vectorial character of the operations ^r-, -^r-, yr-
dx dy dz
makes it often a convenience to represent them by the symbolism
used for vectors. We shall frequently denote them by the symbol
V (pronounced nahla), thus
V- (V. V. V.) - (4 |, I)
and therefore
grad (/> ^ (V.^, V,*^, V.^.) or grad </» = ^^,
and div A ^ V.^4^ + V,^, + V«^. ^ (VA).
The quantity
dx^ "^ dy^ '^' dz^ is called the laplacian of the scalar (/> in honour of the great French mathematician Laplace. Our notation enables us to represent it by V^^*
§2-4] EUCLIDEAN TENSOR ANALYSIS 15
A useful formula, frequently used in electromagnetic theory, is the following : —
div [AB] = (B curl A) - (A curl B) . (2-44) This can be proved by writing out div [AB] in full.
div [AB] = ^[ABL + |[AB], + 1[AB1. The first of the three terms on the right expands to
,dB dA, dB, dA,
and the remaining two terms on the right give jS£ » dA,_dB,_dA,
'dy "^ "^ dy "Sy 'Bf
The pair of terms, marked o, taken together make
5Jcurl^}^. Similarly, we find a pair of terms equivalent to
j5Jcurl^}^, and another equivalent to
J5,{curl^}„ so that six of the terms make up
(B curl A). In the same way the remaining six are seen to make up - (A curl B) and thus the formula is established. An equally important formula is :
curl curl A = grad div A — V^A . . (2'45)
which we can likewise establish by writing out the. left-hand side, or the X component of the left-hand side, in full.
(curl curl A> - ^ 1^^' - ^^4 - ^ i^^" - ^^' icurlcurl A}, -g-|^ -g-| g-|-g- —
{curl curl A}. = ^{-^ + -^) - (-g^^ + ^^), and if we add and subtract -;r— ^ on the right-hand side we get
{curl curl 4}. = - div A - (-^^^ + ^^ + ^), a result which may be expressed in the form (2*45).
or
CHAPTER II
I
THE THEOREMS OF GAUSS, GREEN AND STOKES. FOURIER'S EXPANSION
§ 3. Theorem of Gauss
MAGINE a closed surface, ahc, Fig. 3, and a field vector (A^, Ay, A^) which varies continuously throughout the volume enclosed by it. We shall investigate the integral
\\\
div A dxclydz
(3)
It is important to grasp the precise meaning of this integral.
We suppose the whole volume ahc divided into small elements and each element of volume mul- tiplied by the value of div A at some point within it. The integral (3) is the limit to which the sum of all the products so formed approximates as the ele- ments of volume become indefinitely small. It is not essential that the elements of volume should be rectangular, or that the sum should be expressed by the use of the triple symbol of integra- tion. We may write (3) in the form
FiQ. 3
div A dv
(3-001
where dv represents an element of volume of any shape. It is convenient, however, to use the triple symbol when we wish to draw attention to the 3-dimensional character of the region over which the integration extends.
From the definition of div A (§ 2-4) we have for (3)
16
§ 3] THEOREMS OF GAUSS, GREEN AND STOKES 17
HI
w+is' + wl**"' • ■<»»2>
so that it may be treated as the sum of three integrals. We shall begin with
-^dx dy dz, dx ^ '
and carry out the integration or summation over all the elements of volume from 1 to 2 (Fig. 3) in a single narrow vertical column with the uniform horizontal cross-section dy dz . For this restricted volume we have
nr
I -^^^ ^y ^^ = dy dz -^dx
=^dydz{{A,),-(A,),} . . . .(3-003)
where (A^)i and (A^)2 are the values of A^ at the terminal points 1 and 2 respectively, where the vertical column cuts the surface abc. Let the elements of area at the two ends of the vertical column be (dS)i and (dS)2. It is helpful to imagine short perpendiculars erected on the surface at the points 1 and 2 and directed outwards, each perpendicular having a length equal to the area dS of the corresponding element; (see (dS)i and (dS)2 in Fig. 3). These perpendiculars may be regarded as vectors with the absolute values (dS)i and (dS)2. Let ^i and ^2 he the angles between the directions of the vectors (dS)i and (dS)2 respectively and the X axis. We have then
(dS)2 cos <^2 = dy dz, — (dS)i cos (^1 = dy dz, or
(dS^)2=dydz, — (dS^)i =dydz.
Substituting in (3*003) we get
2
dy dz j ^ydx = (A,d8,), + {AjiS,),
1
or, otherwise expressed, the integral
\\\
— — ^ dx dy dz. dx ^ '
when extended over such a vertical column, is equal to the sum of the products A^dS^, where the surface abc is cut by the column. When the integral is extended over the whole volume abCy i.e. over all the vertical columns in it, we get the sum
18 THEORETICAL PHYSICS [Ch. II
of the products AJIS^ for aU the elements of area making up the surface.
Therefore
^^^^^ dx dy dz = ^^AM.,
where the summation on the left extends over the whole volume ahc, and that on the right over the whole surface ahc. Similarly we have,
IIl^' ^-^ dy dz = ^JA,dS„
and I j I ^ tZa; c?!/ ^2 = | | ^2^^2-
Adding these three equations, we get
r r
{AJS^ + AydSy + A,dS,}
If'
or
[[[div A dxdydz = [f(A dS)
(3.01)
where (A dS) on the right hand is the scalar product of the vectors A = (A^,Ay,A^) and dS = {dS^, dSy, dS^). Equation (3'01) expresses the theorem of Gauss.
§ 3-1. Green's Theorem
Let the vector A in (3 '01) have the form
U grad V,
where U and V are scalars, which, with their first and second
differential quotients, are continuous functions of x, y and z
in the volume abc. We thus have
[jfdiv {U grad V) dxdydz = ff(C7 grad F, dS), or {{{w^W dxdydz -{- [[[(grad U, grad V) dxdydz
= \\{U grad V, dS) . (3-1) Interchanging U and F in (3'1), we get I [ [ Vy^U dxdydz + {{{ (grad U, grad F) dx dy dz
= \\{V grad U, dS) . . (3.11)
§ 31] THEOREMS OF GAUSS, GREEN AND STOKES 19 and on subtracting (3*11) from (3'1) we obtain
I [ f {UyW - YTJ^'V) dx dy dz
= {{(U grad V, dS) - {{{V grad U, dS) (3-12)
This result, known as Green's theorem, was published in 1828 by George Green in an epoch-making work entitled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. If we represent distances measured in the direction of an outward normal to the surface abc by the letter n, the normal component of grad V, i.e. the product of grad V and the cosine of the angle between its direction and that of dS or of the normal, is
dv
dn so that (3*12) may be written in the form
\\i
^l-'sl-^ • ■ ■ <™'
{C/y^F - Vy^U}dxdydz
In this equation let the value of U at any point be equal to
- where r is the distance of the point from the origin. If then
the origin is outside the volume abc over which the triple integral is extended, we have from (3*13)
Since V^- can be shown, as follows, to be zero. We have namely r
ii)
_ 1 dr dx r^' dx
2 /ar\2 1 av
therefore ^^, ^,^^^
Now r^ = x^ + y^ + z^,
dr therefore 2r^r- = 2x
) -^.S- • ■ ■''■»'>
or
dx
dr _x
dx T
20
THEORETICAL PHYSICS
[Ch. II
Further
8V
CX'
_ ^^ 1
dr 9V
Substituting these expressions for — and ^-^^^ (3 •141) we get
vx ox
or
Similarly
and
|
\rj 2x^ dx^ y.5 |
||
|
"(i) |
Sx^ 1 |
|
|
dx^ |
y5 ^3. |
|
|
-<) |
32/2 1 |
|
|
dy^ |
J.5 f3' |
|
|
<) _ |
_ 3«2 1 |
On adding the last three equations, we find
dx''
+
dy''
+
3^2
_ 3(0;^ + ^2 _|_ ^2) _ 3_ _ Q^
^5 y 3
Let us apply (3'14) to the case where the volume integration X
Fig. 31
extends over a region like that indicated by the shaded part of the diagram in Fig. 3'1. This region is enclosed between the
§ 3-1] THEOREMS OF GAUSS, GREEN AND STOKES 21
surface abc and the surface of a sphere of small radius, R, having the origin for its centre. The surface integration is now extended over the surface ahc and over the surface of the small sphere as well.
At points on this latter surface
dn dr'
since the direction of the outward normal is exactly opposite to that of r. Similarly
1
'©
'. I
-lUf-^ii;
dn dr B^
Therefore the part of the surface integral of (3* 14) extended over the small sphere may be expressed as
'dS
OT7" OTT"
where __ and V are average values of -^ and V respectively
over the surface of the sphere. This part of the surface integral is therefore equal to
- 4.7zB-^ - 471 F,
dV —
and since -^ and V are continuous it will approach the
limit
- 47rFo as B approaches zero, if Vq is the value of V at the origin. We have therefore
dl
- F-^ \dS . . , (3-15) dn )
In this formula the surface integral is extended over the outer surface ahc of Fig. 3-1, and it is understood that the volume integral now means not merely the result of integrating over the
22 THEORETICAL PHYSICS [Ch. II
shaded volume, but the limit approached by this integral when R approaches the limit zero.
Imagine the surface ahc to be enlarged, so that the distance r of any point on it from the origin approaches infinity, or so that
- approaches zero ; then it may happen that the surface integral also approaches zero in the limit. It is easily seen that this must happen when V diminishes in the same way as - at great dis- tances from the origin, that is to say, when the product rV never exceeds some finite number, however great r may be. For
'©
- -— and V_SLL are both of the order of magnitude of —, r en ^n ^
whereas the area of the surface is of the order of r"^. In such a case (3*15) becomes
where the integration is extended over all space.
§ 3-2. Extensions of the Theorems of Gauss and Green If A, B and C are three vectors, the quantity
c^A + c^ji, + CAA
is the X component of a vector, since
OA + C,B, + CA = (CB) is a scalar or invariant quantity. Now A^B^, A^By and A^B^ are the XX, XY and XZ components of a tensor of the second rank and it follows that
^x-^ XX I ^y-'- XV ~r ^z-^ XZ
is also the X component of a vector, if G is any vector and T^y, etc. any tensor of the second rank. This will be understood when it is remembered that the components of tensors are defined mathematically by their transformation properties (see §§ 2*2, 2*3). We may similarly infer that
y x^ XX ~r y y-^ XV "T" y «-* xz^
dT dT dT
or ^^ _j_ XV I ^^ XZ
dx dy dz
is the X component of a vector and
dT dT dT
^-^ yx I ^-^ yy i ^^ yz
dx dy dz
§3-3] THEOREMS OF GAUSS, GREEN AND STOKES 23
and -^ + -^ + -^
ox oy dz
are respectively the Y and Z components of the same vector. It is usual to extend the scope of the term divergence to include this vector. Therefore
div T ^ (^-^ ^^J:^ + Em,Em + ^lm +^J^, \ dx By dz ' dx dy dz
dT^^ , dT^,, , dT,
zx I ^^ zy
+ ^^ +
?).... (3-2)
dx dy dz
The method employed to deduce the theorem of Gauss can be applied to prove the statement : —
ill
^+^"+^[c?x%(^«
^^{T^dS^ + T^dS^ + TJS,} . . (3-21
Green's theorem, and the formulae deduced from it, naturally admit of a similar extension. We can, for example, deduce the equation
F^==-^^^^^'dxdydz . . . (3-22
which corresponds to (3*1 6) and in which F^ means the value of F^ at the point r = 0. The validity of this formula is subject of course to conditions strictly analogous to those which apply in the case of (3'16).
§ 3-3. Theorem of Stokes
It has been shown already (2*42) that div curl A is identically zero and therefore
III
div curl A dx dy dz = 0,
the integration being extended over any volume within which the vector A and its first derivatives are continuous. Now applying the theorem of Gauss we get
II
(curl A, dS) = 0 . . . . (3-3)
the integration being now extended over the bounding surface (abed J Fig. 3-3). Imagine the surface abed to be divided into 3
24
THEORETICAL PHYSICS
[Ch. II
two parts by the closed loop a^yd. Equation (3*3) may be written
[[(curl A, dS) + jl (curl A, dS) = 0 . (3-301)
ab c d
Where ah indicates the part of the integral over the portion ab of the surface to the left of the loop a^yd and cd the part extended over the portion to the right of the loop. Now suppose
Fig. 3-3
Fig. 3-31
W
the surface cd to be replaced by another surface ej with the same boundary line apyd. We shaU have
[[(curl A, dS) + [[(curl A, dS) = 0 (3-302)
ah ef
From (3-301) and (3-302) we have
(curl A,dS) = [[(curl A4S),
c d ef
and therefore the value of the integral can only depend on the values of the vector A along the curve a^yd which forms the boundary of the surface cd or ef. This suggests the problem
of expressing (curl A,dS) in terms of what is given for
points on the boundary a^yd (Fig. 3-31) of the surface. Let us construct two sets of lines on the surface, each set containing an infinite number of lines. The first set, which we shall call the d lines, all begin at a common point (1 in Fig. 3-31) and all end at a common point (2 in Fig. 3-31). We shall suppose them to be sensibly parallel to one another in any small neigh- bourhood. The element of area between any two adjacent d lines may be called a d area. The second set of lines, which may be termed 5 lines, are so drawn as to divide the d areas
§ 3-3] THEOREMS OF GAUSS, GREEN AND STOKES 25
into infinitesimal parallelograms, the area of any one of which
may be symbolized by dS. The increment of any quantity
</>, as we travel along a d line in the direction 1 to 2 (shown
in Fig. 3-31 by an arrow), from one 8 line to the next, will be
represented by dcj). In the same way dcj) will represent the
increment of ^ which occurs in travelling along a 5 line, in the
direction indicated by the
arrow, from one d line to the
next. If the letter I be used for
distances measured along any
of these lines, an element of
area dS will be equal to
61. dl. sin d (see Fig. 3-32). As
usual we shall regard dS as a
vector and write
dS = [61, dl] We can visualize dS as a short displacement perpendicular to the surface of the element and directed away from the reader. We evidently have
dS^ — dy dz — dz dyA
dSy = dz dx — dx dz,> (3'31)
dS^ = dxdy — dy dx.)
In these equations
51 = (dx,dy,dz)\ /a.crtn
dl = (dx,dy,dz)f [6 6ll)
Let the X co-ordinate of the point 1 (Fig. 3-32) be x ; the X co-ordinate of the point 3 will be a; + dx, and as we pass from
3 to 4 we realize that the X co-ordinate of the point 4 must be
X -}- dx -{■ d{x + dx), or x -{- dx -\- dx + ddx .... (3'312)
If we travel from the point 1 to the point 4 by way of the point 2, we find the X co-ordinate of 2 to be a; -f dx and that of
4 to be
X -{- dx -\- d(x -\- dx), or X -\- dx -{- dx -{- ddx .... (3*313)
Both of the expressions (3'312) and (3*313) represent the X co-ordinate of the same point, and it follows that
ddx = ddx (3-314)
This means that the operations d and d are interchangeable, at any rate when applied to the co-ordinates.
The integral (curl A,dS) over a surface bounded by
28
THEORETICAL PHYSICS
[Ch. II
the line a^yd may be treated as a sum of integrals, each ex- tended over a d area. A typical d area is shown in Fig. 3-33.
The surface integral over it may
be written
J {[curl AldS, + [curl AldSy +
[curl^L^^J,
in which the double symbol of integration has been dropped, since we are now dealing with the sum of a singly infinite set of elements extending over the d area from 1 to 2. Suppose the integrand to be written out in fuU, using the definition of curl in § 2-4 and equations (3'31). The part of the integral involving A^ only is
2
-^((32 dx — dx dz) — -~\dx dy — dy dx) ,
Fig. 3-33
J [11% + 'H"
dy
Adding and subtracting
-^dx dx, ox
+
-^dz\dx . dz j J
we get for this part of the integral
[SA^dx — dA^dx}.
Now add to this result the integral
2
I d{A,dx)
■which is equal to zero, since dx vanishes at the points 1 and 2.
§3-3] THEOREMS OF GAUSS, GREEN AND STOKES 27 We have therefore for the part of the integral under investigation,
2
[SA^dx + d{AJx) — dAJx] 1
2
= j [dAJx + A^ddx], 1
2
= j [dA,dx + ^,^6^a^], 1
2
1 Now this expression is the difference of two line integrals, namely
2 2
AJ.X - A^dx,
LI Rl
the one distinguished by the letter L being taken along the left- hand boundary of the d area and the other, marked R, along the right-hand boundary. The difference can be expressed as the line integral
1
AJx,
taken right round the d area in a clockwise sense.
When we take terms involving Ay and A^ into account we
find the surface integral j I (curl A,dS) over the d area to
be equivalent to the line integral (A^dx + Aydy + A^dz) taken round it. Otherwise expressed
jj(curl A,dS) = j (A,dl),
dl being a vectorial element of length along the boundary of the d area.
Finally the integral (curl A,dS), when extended over
the whole surface, is equivalent to the sum of all the line
integrals, (A,dl), taken round all the d areas of which the
surface is made up. This sum must be equal to the line integral
28 THEORETICAL PHYSICS [Ch. II
taken round the boundary apyd, since along every other line numerically equal integrals are extended in opposite senses. We thus arrive at the important result known as the theorem of Stokes,
^(Adl) = [[(curl A,dS)
(3-32)
The line integral is extended, as the symbol ^ is meant to indicate, round the boundary a^yd of the surface over which
the integral I is extended.
itegral I i
If we imagine a screw turning in the sense in which the line integral is taken round the boundary, the vectorial elements of area dS will be directed to the side towards which the screw is travelling.
§4. Fourier's Expansion
A very extensive class of functions can be represented, for a limited range of values of the independent variable, by the sum of a series of trigonometrical terms. If cj) be the independent variable and /(</>) the function, we have f((f>) == ^0 + ^1 cos <^ + ^2 cos 20 4- ^3 cos 3^ + . . .
+ jBi sin ^ + ^2 sin 2(/> + JSg sin 30 + . . . (4)
The coefficients are defined by the equations,
i^\mdr,
+ n
A^^ =- f{r) cos ^T cZr, >^ = + 1 to + 00 . (4*01)
+ n
B^ —- \ /(r) sin nr dr
n J
where n may have all integral values from 1 to oo. The sum of the series (4) will correctly represent the function /(0), subject to a qualification given below, for all values of 0 between — 7t and + 71. The expansion is due to Jean Baptiste Fourier and wiU be found in his TMorie Analytique de la Chaleur published in 1822. Its validity was established by Lejeune Dirichlet in 1837 for aU one- valued functions of the type which can be represented graphically. A discussion of the validity of Fourier's expansion (4) is beyond the scope of this book ; but if we accept
§4] THEOREMS OF GAUSS, GREEN AND STOKES 29
the validity, it is easy to prove that the coefficients are those defined by (4*01). In evaluating any integral, such as
+^
/(t) cos {nr)dr,
— IT
we have simply to make use of the relations : COS 2 (nr) dr — 71,
+ ?r
sin^ (nx) dx = n,
+ rr
J
COS (nr) sin (mr) dr = 0,
n — m, or n j^ m,
(4-02)
+ 77
COS (nr) cos (mr) dr =0, n ^ m,
+ T
sin (nr) sin (mr) dr = 0, n ^ m.
In Fig. 4 the abscissae are the values of the independent variable, ^, and the ordinates those of a function, /(</>), which
-37r
^ ^
Fig. 4
may be given quite arbitrarily. Whether the function is periodic or not, the expansion will represent it correctly between — tz
30 THEORETICAL PHYSICS [Ch. II
and + n, with the possible exception of the points — n and + 71 themselves. Outside this range of values of the independent variable, it is evident that the arbitrarily given function /(<^) cannot in general be equivalent to the sum of the series (4), since the periodic character of the cosine and sine terms neces- sitates that the values of the function in the interval — n to + n will be reproduced by the expansion in every further interval of 271, e.g. from 7i to Stt, or from — Stt to — n. This is indi- cated in the figure by the broken lines.
We notice that the sum of the series (4) may approach two different limiting values from the two sides of the points —n and 4- 7i, and it can be shown that the result obtained by sub- stituting — jr or + TT f or (^ is the arithmetic mean of the two limiting values. If we wish to expand an arbitrary function \p{x) in a series of cosine and sine terms which wiU be valid for any prescribed range of values of x, we can quite simply reduce the problem to the one we have discussed by introducing a variable
, 71
and we shall arrive at a result which is valid for values of x between — L and -\- L.
When the expressions (4'01) for the coefficients A and B are substituted in (4), Fourier's series takes the form
/(^) = — f{r)dr + - y^ /(r) cos nr cos ncjxir
— IT n=+l — XT
4- - y f(r) sin nr sin n(j)dr,
or
W= + l — TT + 1T n=+QO +77
— 77 n = + l —77
Since
cos X = cos (— x) we may evidently write (4*03 ) in the alternative form
+ 77 n= — 00 +77
§4] THEOREMS OF GAUSS, GREEN AND STOKES 31 and on adding (4*03 ) and (4*031) we get
+ 1T «= + ltO+00 +77
m4') =l\ mdr +1 2^ J fir) cos n{r - 0)rfr,
— 77 n=— ItO— 00 —77
and hence
n= + CO +77
M)=-^ Z j/Wcos7i(T-c^Mr. . (4-04)
r by and (j) by
n= — oo —IT
If now in this formula we replace
(7
» a
1 a
where a is a positive constant, we shall have
/(r) cos n(r — (b)dr =/( - ) cos -{a — w)da.-.
\a/ a a
Write - = A, and suppose a to be very large, approaching oo
in the limit. Then - becomes dX and the summation, 2", with a
respect to n becomes an integration, , with respect to X.
Therefore, if we write F{a) for /(-) and F{xp) for /(- ), and
observe that the limits for a = ar must be — oo and + oo, we arrive at the interesting result
+ 00 +00
F{^^)) = ^ [ [ F{a) cos A((7 - tp)dadX . . (4-05)
— 00 —00
which is known as Fourier's Theorem.
The derivation just given is not rigorous, but it shows the connexion between the theorem and Fourier's expansion. It can be proved to be valid for arbitrary functions of the type that
+ 00
can be exhibited graphically, provided the integral F(a)da is
— 00
convergent. The arbitrarily given function F(\p) may have a finite number of discontinuities, i.e. there may be a finite number of values of ^ at each of which F(\p) has two limiting values. For these values of \p the integral in (4'05) gives the mean of
32 THEORETICAL PHYSICS [Ch. II
the two limiting values and in this respect is like Fourier's expansion.
§ 4-1. Examples of Fouribii Expansions
When/(^) in (4) is equal to ^, all the coefficients, A, vanish, since the value of the function merely changes in sign when the sign of ^ is changed. Therefore
M) =4>= ^^n sin ^^.
By (4-01)
+ 7r
'■'^l\
T sin nr dr.
This gives on integrating by parts,
j5„ = — - cos nn, n
and therefore
^ = 2{sin ^ - J sin 2^ + i sin 3(?f» h . • .} . (4-1)
The sum of the series (4*1) approaches the limit tt as ^ approaches n from below, and the limit — tt as 0 approaches — n from above. On account of the periodicity of the terms it will also approach the limit — tt as </> approaches n from
fi^) -^
Fig. 4-1
above. There are therefore two limiting values of the sum of the series at n and at every point nn where n is odd. When however we substitute nn for </> the sum of the series is found to be zero, which is the arithmetic mean of — tt and + ^ (Fig. 4-1 ).
§4-1] THEOREMS OF GAUSS, GREEN AND STOKES 33
If we wish to represent ip{x) — x by a trigonometrical series in an arbitrarily given interval, e.g.
X
we may substitute tt- for <^ in (4'1) thus
nx ^( . nx 1 . ^nx , 1 . Stzx
_ = 2jsm-^ --sin 2^ + _ sm — - + . .
and so we get
2L( . Jtx 1 . ^Ttx . 1 . ^nx . I /A ii\
When
and
the function is an even one and the coefficients B vanish (Fig. 4-11). Therefore
f(<t>) =Ao + ZA,, cos nct>. 1 Equations (4*01) give us
or and
^0 -2'
^^ = -j — r COS nrdr ■}- \ r cos nt dr
Integrating by parts, we find
2
A,::^~{cosn7t-l),
34
Therefore
and
THEORETICAL PHYSICS
[Ch. II
= 0,
A,= - A,= -
and so on.
2 "
M)
4 4
¥^'
4
Therefore -jcos <J^ + 3-2 cos 3^ + — cos 5^ -f . . .} (4-12)
The sums of the series (4-1) and (4-12) are equal when 0 ^ ^ <[ itt.
0 If
f{cl>) = -l, ^<0,
Fig. 4-12 (Fig. 4-12) we find the expansion
sin
If
and
(Fig. 4-13)
M)-
<j> + -sin B<j> + -sin B<j> + . . .1
im = - 1,
(4-13)
^ 2'
^>I'
i'-l<^<i'
we obtain
4(1 1 = -j cos 0 — - COS 3^ + - cos 50 h
(4.14)
7p 0
7T
Fig. 4-13
§41] THEOREMS OF GAUSS, GREEN AND STOKES 35
As a further example take the case, illustrated in Fig. 4-14, where
0
Fio. 4-14
M) =n -4>,
m = - (TT + 4,),
(4.15)
The function is an odd one and hence the coefficients A are all zero, while
Bn = -V- sm n-,
so that the appropriate expansion is
4 f 1 1
/(</>) = -| sin ^ - 3-2 sin 3</) + - sin 5^ - + .
As a concluding example we may take the function /((/,) = — sin (/>, —n <,(l)<0, M) = + sin 9f>, O^cj^^n,
(see Fig. 4-15), for which we find the expansion 2 _ 4|cos 2(f) cos 4^ cos 6(^ ^ S rT3~" 3-5 5-7 '*
/(^)
(4.16)
Fig. 4-15
In all the examples where points of discontinuity occur, e.g. at ^ = 0 in (4.13) and at ^ = - ^ and ^ = + - in (4.14), it
36 THEORETICAL PHYSICS [Ch. II
may be verified that at such points the sum of the series, which is of course a one-valued thing, is equal to the mean of the two limiting values of /((/>) at the point in question.
We obtain interesting verijfications of the formulae given above when we substitute special numerical values for </>. For
example, if we substitute the value - for ^ in (4'1) or in (4* 13),
or if we substitute the value 0 in (4*14) we get
- = 1-1 + 1-1 + -...
4 3 5 7
If we put ^ = 0 in (4'12), or ^ = - in (4«15) we obtain
-^ = 1+1 + 1+1+...
Both of these formulae are well known, and can easily be arrived at in other ways.
It is possible, and often convenient, to give the expansion (4) the form
v=+co
m = Z ''^'"' • • • • (4-17)
!/ = — 00
This is effected by the substitutions
cos n(j) sin n(^
2
2i '
where i is the usual abbreviation for V — 1. The series (4'1), for example, when expressed in this way, becomes
^ ^2 3
_!_ i e-i4> — ie-i20 _J_ l.Q-iS'i^ h . . .
2 3
§ 4-2. Orthogonal Functions
The most important property of the trigonometrical functions in a Fourier series is that which finds its expression in equations (4*02 ). An aggregate of functions like cos nr and sin nr with this property is called a system of orthogonal functions, because of the analogy between the equations (4'02) and the equations (2*23) and (2*24) which give the relations between
§4-2] THEOREMS OF GAUSS, GREEN AND STOKES 37
the direction cosines of mutually perpendicular lines. The analogy becomes more obvious if we adopt the functions
cos {m) , sin (nr)
Vn Vn
instead of cos (nr) and sin (nr), for then the non-vanishing integrals of (4*02) are
When the non- vanishing integrals are thus modified so as to have the value unity, the orthogonal functions are said to be normed. For the interval — n to + jt the normed trigono- metrical orthogonal functions are
cos (n^ ^^^ nm (nj,) ^ n=^\,^,...oo
Vn V,
7t
and to these we may add the constant ^ , since
V27t
— TT
In terms of the normed functions, the Fourier expansion becomes ^/JLN / 1 \ I cos cf) cos 20 ,
+ ^,E2i + ^,EL|^+ ... (4-2)
the coefficients being given by
"^^iHvEy^'
— TT
71 = 1, 2, 3 ... 00.
^„= j/(.)j!iL|i)j^.,
38 THEORETICAL PHYSICS [Ch. II
Let us now introduce the notation,
V2n Vn
E^i^) = £2!i, EM) = !HLP,
Vn V
n
^ J cos (m^) „ sin(m + l)<^
Vn Vn
The Fourier expansion now becomes
m=Zc,EM) .... (4-21)
0
and since the integral relations (4*02) now have the form j E^(r)EJr)dr = I, n = m,
= 0, 71=^ m . . . (4-22) the coefficients c^ are given by
+ 77
On = ^^nr)E,(r)dr, n = 0, I, 2 . . . oo . (4-23)
— TT
Systems of orthogonal functions are of course not limited to trigonometrical functions. A system of orthogonal functions -^71 (^) by nieans of which an arbitrary function /(^) can be expanded in the form (4'21) is called a complete system of orthogonal functions.
CHAPTER III INTRODUCTION TO DYNAMICS
§ 5. Force, Mass, Newton's Laws
THE notion of force has its origin in the feeling of muscular effort . Quantitative estimates based immediately on the feeling or sensation of effort are, however, too rough and uncertain to serve any purposes where precision and consistency are demanded. Consequently force, like all other physical quantities, is measured by devices which entirely eliminate any dependence on the intensity of a sensation. Typical of such devices is a spring (as in a spring balance, for example). Instead of the uncertain comparison of two weights by feeling how big are the muscular efforts exerted in supporting them, it is better to use the extensions they produce in a spring as measures of their weights. The procedure in measuring temperatures is quite analogous. The sensation of warmth i' or hotness plays no part whatever in such measurements. Let us examine more closely the measurement of force by the extension of a spring. To fi:K our ideas we may continue to keep the spring balance in mind. The upper end. A, of the spring is attached to a support, which for our i Y purposes may be supposed to be rigidly fixed. A heavy Fig. 5 body is suspended at the lower end, B. The extension can be represented completely by a vertical line (XY in Fig. 5) the length of which is made equal to it. The upward and downward directions in XY have equal claims on our atten- tion, and we shall say that a force is exerted in an upward direction on the body suspended at B, and that a force is exerted in a downward direction on the support at A. Both forces are measured by the extension, XY, of the spring and are therefore numerically equal to one another. This is the (so-called) law of action and reaction, which here emerges as a necessary consequence of the measuring device and associ- ated definitions. It is implied, in what has been said, that the unit force is the force associated with the unit exten- sion, and it follows that different units of length will, in 4 39
4:0 THEORETICAL PHYSICS [Ch. Ill
general, have to be used with different springs. There is no difficulty in deciding when the extensions produced in a number of springs are aU associated with the same force ; but the fol- lowing circumstance has to be considered : though different springs may agree with one another approximately, or even very closely, when used to measure forces extending over a certain limited range, and though the intervals on the evenly-divided scales attached to them may have been chosen so that all the springs are in precise agreement in the case of one particular force (the adopted unit) ; they wiU nevertheless be found to dis- agree to some extent when measuring other forces. In fact, when springs are used in the way described, each measures forces according to a scale of its own. There is a close parallel to this in the measurement of temperature by thermometers of different types. We find it necessary to define a force scale independent of the peculiarities of the particular measuring device — some- thing precisely similar is done in the measurement of temperature. We might do this by adjusting a number of weights so that they all produce the same extension in one particular spring, when hung from it separately. We should then be able to calibrate any spring by suspending the weights from it, one, two or more at a time. The following observational facts point out another way of defining a force scale (which will, in fact, amount to the same thing in the end) : In the first place, if the point of sup- port, A, referred to above, be caused to ascend with a uniform speed, the extension of the spring (after the initial oscillations have been damped out) will not be altered. This means that no force is needed to maintain the constant velocity of the body at B. In the next place, if A ascends with a constant acceleration, the extension of the spring will be increased by a definite amount, proportional (approximately) to the acceleration. In other words, the upward acceleration of the body suspended at B is approximately proportional to the resultant force (as measured by the spring) to which the acceleration is due. In consequence of these facts it is possible, and many reasons make it desirable, to define the force scale by the statement
F = ma (5)
where F is the resultant force, a is the acceleration, and m is a constant characteristic of the particular body, and called its mass. If we make m unity for some arbitrarily chosen body, we shall thereby fix the unit of force at the same time. It will in fact be the force which causes it to move with the unit ac- celeration. In the case of bodies made entirely of the same material, e.g. brass, the mass is found to be very nearly propor-
§5] INTRODUCTION TO DYNAMICS 41
tional to the volume. This is the justification of the rather imperfect definition of the mass of a body as the quantity of material in it.
Representing the velocity of the body by v, equation (5) may be written
or F = _(mv),
(XT/
or finaUy ^^^ (^'^^^
where M = mY ...... (5*011)
is called the momentum of the body. Equation (5*01) embodies in a single statement Newton's first and second laws of motion ; the law of action and reaction being Newton's third law. In applying these laws generally, we must regard the body as very small in its dimensions, i.e. as a particle, in order to avoid the difficulty which would appear if the velocities or accelerations of its parts differed from one another ; and in dealing with classical dynamics we shall assume, as Newton appears to have done, that the mutual forces, exerted by two particles on one another, are directed along the straight line joining them.
The unit of mass adopted for scientific purposes is the gram, i.e. the mass of a cubic centimetre of water at the temperature of its maximum density ; the unit of length is the centimetre, and the unit of time the mean solar second . With these funda- mental units, the unit of force fixed by (5) or (5*01), i.e. the force causing the unit rate of change of momentum, is called the dyne.
It is an experimental fact of great importance for the science of physics that unsupported bodies at the same place, i.e. bodies which have been projected and are falling freely, have the same downward acceleration. This is usually represented by the letter g, and is equal to 980-6 cm. sec."^ in latitude 45°, and varies from 978 cm. sec.~2 at the equator to 983-4 cm. sec."^ at the poles. By ' freely falling ' body is to be understood of course one which is not subject to the resistance of the air or any other sort of interference. There is therefore a downward force acting on the body, equal to mg. This is its weight. At the same place, therefore, the weights of bodies are proportional to their masses ; but whereas the mass of a body is a constant characteristic of it,^ and independent of its geographical position, its weight will vary
1 This statement will be modified when the theory of relativity is described.
42 THEORETICAL PHYSICS [Ch. Ill
with the latitude in consequence of the variation of g. The acceleration, g, is the weight or gravitational force jper unit mass and we shall term it shortly the intensity of gravity. This is, for several reasons, preferable to the ambiguous term ' accelera- tion due to gravity '.
§ 5-1. Work and Energy
The scalar product (Fdl), or FJtx + F^dy + F^dz, where p = (i^^^ Fy, jPJ is any force and dl = {dx, dy, dz) is a small displacement of its point of application, is called the work done by the force during the displacement dl. And when the point of application of the force travels from any point A along some
path ABC to another point C (Fig. 5-1), ^ ^^.^ — j^ the work done may be represented by
f(Fdl).
ABC
It may happen (and this is a very important case) that the work done is independent of the path. Starting -pjf.^ 5.1 from some fixed point A, the work done
will depend only on the position of C. If we represent the work by W, we have W = function {x,y,z), where x, y, z are the co-ordinates of C. And if we take some neighbouring point C with co-ordinates {x -{- dx, y -f dy, z + dz) the work done will be W -{- dW, where
dW = — dx -{- — dy -f — dz . . (5*1)
dx dy ^ dz ^
so that
_eW ^ _dW J, _dW ^ dx' ' dy' ' dz' We may therefore say that, when the work done is indepen- dent of the path, the force is the gradient of a scalar quantity. Conversely, when the force is the gradient of a scalar quantity at all points in a certain region, the work done between two points A and C in the region is independent of the path, since the curl of a gradient is zero (§ 2-4) and therefore
[|(curlF,dS)
and consequently, by the theorem of Stokes, the integral
ct (Fdl)
taken round any closed loop ABCDA in the region will be zero.
§51] INTRODUCTION TO DYNAMICS 43
If the work done depends on the path, then the integral ^ {Fdl) round a closed loop such as ABCDA will, in general, differ from zero ; and since
(f)(Fdl) = [[(curl F, dS),
we see that curl F cannot be everywhere zero. Therefore F cannot be represented by a gradient everywhere. When dW is expressed by formula (5*1) it is called a complete or perfect differential.
If we adopt the centimetre, gram and second as funda- mental units for length, mass and time respectively, the unit of work derived from them is termed the dyne -centimetre, or more usually the erg. It is the work done by a force of one dyne when the point where it is applied moves one centimetre in the direction of the force.
Let us consider the motion of a particle under the influence of a force F = {F^, Fy, F^). By equation (5) (Newton's second law) we have
d'^x ^
-S=^^ (5-")
dx dij dz
Multiplying these equations by — , -—- and — respectively we get
CtZ 0/1/ Cf/v
dx d^x m d { /dx\ ^) ri dx
dt dt^ 2 dt(\dtj j ""dt'
dt dt^ 2 dt[\dtj J 'dt'
dz d^z _md(/dz\^) _-n,dz^ '^tltt^ ~'2Jt\\dt) ) ~ 'dt'' and, on adding, we have
dt[2 j ""dt ~^ 'dt "*" 'dt
where v is the velocity of the particle. Therefore
— V^ — — Vo^ =
2 2 °
m „ m
_V2 — _
2 2
or — v^ — — Vo^
^{FJx+Fydy+FM = J(Fdl) (5-12)
44 THEORETICAL PHYSICS [Ch. Ill
In this equation v and Vq mean the velocities of the particle at the end and at the beginning of the path over which the integration is extended.
When the force is the gradient of a scalar W the integral on the right of (5" 12) has the form
J 1 ao; ^ dy ^ ^ dz I
= W - Tf 0, and therefore equation (5' 12) becomes
^ -W = '^-W,. . . .(5-121)
If we replace W hj — V and use the letter T for , we have
T + F = To + Fo . . . . (5-122)
The quantity T + F remains constant. This quantity is called the energy of the particle, T being its kinetic energy, a function of its mass and velocity, and F the potential energy, a function of its position. Equation (5*122) affirms that the energy of the particle remains constant.
The conception of energy will be developed more fully in subsequent chapters ; for the present we are concerned only with the two kinds of energy which have just been described, and it will be noted that energy is measured in terms of the same unit as work. We shall normally regard the centimetre, gram, and second as our fundamental units — the question as to whether three fundamental units will suffice may be deferred till later — but it is obvious that the foregoing formulae (5*11) e^ <scg. are quite independent of the particular choice we make of fundamental units.
§ 5-2. Centre of Mass
Imagine a number of particles, the masses of which are mi, m2, mg, . . . m^, and their co-ordinates {x-^,y-^,z-^, {X2,y2,^2), (a; 3, 2/3, 23), ... (Xg, 2/s, Zg) respectively. The centre of mass (x, y, z) of the system of particles is defined by
Mx = HrrigXg,
My = Em,y, (5-2)
Mz = Em^Zg, where M = Em^,
the summation being extended over all the particles of the
§5-3] INTRODUCTION TO DYNAMICS 45
system. Let F^ be the sum of all the X components of the forces exerted on the particles of the system ; we shall have
Em^ =^F
and two corresponding equations for Fy and F^ ; or by (5 '2)
M— Lf
^^.-Fy (5-21)
M^ =F
dt^ ''
Therefore the motion of the centre of mass is the same as it would be if aU the masses were concentrated in it and aU the forces applied there. In equations (5*21 ) we may regard F as the resultant of all the forces of external origin, since by Newton's third law, those of internal origin will annul one another. An important case is that in which the system is free from external forces. In this case we can infer from (5 •21) that the centre of mass will move with a constant velocity — which may of course be zero. It is immaterial whether the particles con- stitute a rigid body or not.
§ 5-3. Path of Projectile
Let a particle of mass m be projected from the origin of rect- angular co-ordinates with a given initial velocity, and suppose the X axis directed vertically upwards. If g is the intensity of gravity, i.e. if g is the weight of the unit mass, the equations of motion of the particle are :
dH m^ = - mg,
-S=« (5-^)
The two latter equations (5*3) give us
y = a^t -\- /^i,
z = a,t -{- ^^ (5-31)
where ai, ^i, a^, P^, are constants. If we eliminate t from (5*31) we get the equation
or a^z = a^y + a^p^ — aa/^j .... (5-311)
46 THEORETICAL PHYSICS [Ch. Ill
which is the equation of a vertical plane. The path of the particle therefore lies in a vertical plane, and it is convenient to place our axes of co-ordinates so that this vertical plane is the XY plane, and the Z co-ordinate of the particle is permanently equal to zero. The equations of motion are now
d^x _ di^ ~ ~^
Therefore
^= - igt^ + At + B)
y= at+^j ... (5 ^1^)
If we eliminate t from these equations we obtain the equation of the trajectory of the particle,
« = - xjy^ + A(y-^) +B . . (5-32)
The constants m and g, which appear already in the differential equations (5 '3) before any steps have been taken to integrate them, we shall term inherent constants. Such constants are characteristic of the system to which they belong and are not in any way at our disposal. It is otherwise with the constants of integration A, B, a and ^. If the particle is projected from the origin at the time ^ = 0 with a velocity V and in a direction making an angle d with the horizontal axis, Y, we have from (5*3 12)
B =0,
and since
dx , , A
|
5f = «' |
|||||
|
the |
jrefore |
F sin 0 = A, |
|||
|
F cos 0 = a. |
|||||
|
Substituting |
in (5*32), we have |
||||
|
r — - ir/ ^ -I- |
F |
sin |
e If |
||
|
^Vcos2 0 + |
F |
cos |
6^ |
||
|
or |
2:i:.F2cos |
^0 = -gy^ + 2F2sin |
d |
.cos |
e.y |
|
If |
in (5*321) we |
put a; = 0, we find |
|||
|
or |
y =0 F2 sin 26 y = :: ' |
(5-321)
§5-4]
INTRODUCTION TO DYNAMICS
47
the latter of these values of y representing the horizontal distance travelled by the projectile between two instants when it is in the plane x = 0.
§ 5-4. Motion of a Particle fndee, the Influence of a Central Attracting or Repelling Force
Let the point, towards or away from which the force on the particle is directed, be the origin, 0, of rectangular axes of co- ordinates, and let
r = {x, y, z)
be the displacement of the particle from the origin. Let the absolute value of the force be F. We have for the equations of motion,
X
d^x
F.
dt^ r
(5-4)
dh dt^
F.^.
Fig. 5-4
since F is directed along the radius
vector r. Multiply the first two of
these equations by y and x respectively and subtract.
thus obtain
We
mx
or
m
dt^ d ( Jiy
di
my-
d^x df^
0,
Consequently
/ dy\ d / dx\
\dt) ~ ^tVdt)
= 0.
dy
mx-^ dt
(X/X y-.
where Q^ is a constant. Similarly
dz
dy ^ my— — ms-^ = L?,,, ^dt dt
dx dz ^
mz—- — mx-- = 14. dt dt "
(5.41)
Evidently Q^, Qy, Q^ are the components of a vector. In fact
£1 =m[rv] (5-411)
(§ 2-1) where v is the velocity of the particle. The constant £1 is called the angular momentuin of the particle. Since
48 THEORETICAL PHYSICS [Ch. Ill
the vector product [rv] has the absolute value rv sin 6 (Fig. 5-41), it must be equal to twice the area swept out per unit time by the radius vector r. For if in Fig. 5*41 the points 1 and 2 represent two neighbouring positions of the particle dl apart, the corre- sponding area swept out by the radius vector is J . r sin 0 . dl, and dividing by dt, the time taken by the particle to travel dl, we see that the area swept out per unit time is in fact |r sin 6v.
The angle swept out by the radius vector while the particle travels the distance dl is
dl sin 6
and consequently the angular velocity of the particle is
V sin 6
io = .
r
We may therefore write
^=mr2.to . (5-412)
We may summarize as foUows : —When a particle moves under the influence of a force directed towards or away from a fixed point its angular momentum remains constant.
Since £1 is equal to the vector product of r and v multiplied by the invariant factor m, it must be at right angles to the directions r and v (see § 2-1). Therefore the scalar product (Sir) is equal to zero, that is
Q.^x +Q,yy +Sl,z =0 , . . . (5-42)
This is the equation of a plane passing through the origin, i.e. through the attracting centre.
§ 5-43. Angulab Momentum of a System of Particles free FROM External Forces
We may write the equations of motion of any one of the particles, which we shall distinguish by the subscript, s, in the usual way :
~dP
'""S-J^ = Fsy,
'dt^
§5-43] INTRODUCTION TO DYNAMICS 49
By a procedure similar to that used to deduce equations (5*41 ) we get :
^sik^ - x,^\ = ^sF.. - ^.P., . (5-43)
dt [ ^ dt ^dt
Vs-iT \ — ^s-^ sy Us^ &
'dt[ 'dt ""'dt
which may be expressed more briefly in the following form :
-|n, = [r,FJ .... (5-431)
The vector product [r^Fg] is called the moment of the force F^ with respect to the origin. If we add aU the equations (5*431) for the whole system of particles, we have
|sa=S[r,FJ (5-432)
the summation being a vectorial one ; i.e.
Sa, ^ 0Q,,, Si3,„, Si?,,),
|[r.F.] - {^(VsFs. - 2.-f»)> S(z,^,, - x,FJ, ^{x,F,„ - y.FJ-^
Equation (5*432) states that the rate of increase of the result- ant angular momentum about the origin is equal to the resultant of the moments of the forces with respect to the origin. Now there are by hypothesis no forces of external origin, and if we suppose that the mutual action between any two particles is directed along the straight line joining them, the right-hand side of (5*432) can be shown to be zero. For the moment of a force is numerically equal to the product of its absolute value and the perpendicular distance from the origin to its line of action. Therefore the resultant of the moments due to the mutual action of any two particles on one another must vanish, since the forces are equal and opposite and the perpendicular distance mentioned above is the same for both.
We conclude therefore that the resultant angular mo- mentum of a system of particles remains constant in magnitude and direction provided the only forces are those due to the mutual action of the particles on one another and that the force exerted by any particle on another is directed along the straight line joining them.
50 THEORETICAL PHYSICS [Ch. Ill
§ 5-5. Planetary Motion
We shall now study the motion of a particle under the influ- ence of a central force varying inversely as the square of the distance of the particle from a fixed point, which we shall take as the origin of a system of rectangular co-ordinates. Since, as we have already proved (§ 5-4), the particle moves in a plane, we shall place our co-ordinates so that this plane is the XY plane. The Z co-ordinate of the particle will remain constant and equal to zero. We now introduce polar co-ordinates, r and 6, defined by
X = r cos 6,
2/ = r sin 6,
The kinetic energy of the particle is easily calculated. Since if dl is an element of the path of the particle, dl^ = dx^ + dy^
= (dr. cos 0 - r sin d.dd)^ + {dr.Qin 6 -\- r cos d.dd)^
= dr^ + rHd^
Therefore the kinetic energy
©■=KJ)'+Kf)' ■ ■ <'■"
rlfl
We may eliminate the angular velocity, — , by making use of
do
"="*<!) • • • • ^'-'''^
The kinetic energy may thus be expressed in the form
;m\
, /dr
^ \dt
dr\^ Q'
The central force is expressed by
where J5 is a constant, positive or negative, according as it is a repulsion or attraction.
The X. component of the force will therefore be
F =?^
-■ = -£©•
p
Therefore -, plus a constant, is the potential energy of the
§5-5] INTRODUCTION TO DYNAMICS 51
particle (see § 5-1). It is convenient to fix the arbitrary constant in such a way as to make the potential energy zero when the particle is infinitely distant from the centre of force. Repre- senting the sum of the kinetic and potential energies by E, we have
. /dry ^ Q^ ^ B ^ ,- -,,
In this equation Q and E are constants of integration (§§ 5-1 and 5-4). To get the equation of the path or orbit of the particle we eliminate dt in (5*51) by means of the equation of angular momentum (5*501) which may be expressed in the form,
dt = -jr-dd.
Equation (5*51) now becomes
J^(±y+J^+B^E . . .(5-511) 2mr^\dd/ 2mr^ r
We now introduce a new variable u defined by
u=- (5-512)
r
so that dr = — —^ (5*513)
Substituting in equation (5*511) we get
Differentiating this equation with respect to d and dividing out
the common factor 2-— we obtain dd
d^u . mB ., _^,
5e-. + « = -^ • • • • (5-52)
The general solution of (5-52) is
■» = - ^ + -B cos (e - ^) . . . (5-53)
where R and r} are constants of integration. This equation may be expressed in the form :
r^ ^^^ . . . .(5*531)
1 + £ cos (0 - ?^) ^ ^
where £ and r\ are constants of integration. The latter evidently depends on the fixed direction, that of the x axis, from which
52 THEORETICAL PHYSICS [Ch. Ill
the angle 6 is measured and it is convenient to choose our system of reference so that ^ = 0. There is no loss of generality in taking £ to be positive only, since the effect of changing the sign of e is just the same as that of rotating the fixed direction, from which 6 is measured, through the angle n. Equation (5*531 ) represents a conic section ; hyperbola, parabola or ellipse. If the central force is one of repulsion, B is positive and therefore the numerator of (5*531) is negative ; and since only positive values of r have a physical significance, it follows that the de- nominator must also be negative. This is only possible when s is greater than unity. The orbit of the particle is therefore one branch of a hyperbola (represented by the full line in Fig. 5-5).
Fig. 5-5
The centre of force is the focus 0. It will be seen that the path of the particle goes round the other focus, 0'. The asymptotes make angles Oq with the axis of reference (the X axis) determined by the equation
. 1
cos Oq = — -. e
If the central force is one of attraction (B negative), the numerator of (5*531) is now positive and consequently the de- nominator must be positive too, in order to furnish physically significant values of r. It is obvious that s is not now restricted to values greater than unity ; but if it should be greater than unity, we have again a hyperbola (Fig. 5-51 ) and the same relation,
cos do = — -, £
for the directions of its asymptotes as in the case illustrated in Fig. 5- 5. In the present case, it will be observed, the orbit goes round the focus 0, the centre of force. When £ == 1, the path of the particle becomes a parabola, the attracting centre being in the focus.
§5-5] INTRODUCTION TO DYNAMICS 53
The case where e is less than unity is of special interest. The orbit is now elliptic, the centre of force being in one of the foci. If a and h are the semi major and minor axes respectively of the ellipse, we have
-^ =- .... (5-532) mB a
The force exerted on one another by two gravitating particles is
-^ (5-533)
where M and m are the masses of the particles, r the distance between them and 0 is the constant of gravitation. If the ratio
— is negligibly small compared with unity, the centre of mass
of the pair of particles practically coincides with the particle M
O'
>x
Fig. 5-51
and if we use axes of co-ordinates with the origin in the centre of mass we shall have the centre of attraction fixed in the origin.
The force exerted on m is — where
B= - OMm .... (5-534) by (5-533). It follows from (5-532) that
QM^^^Il .... (5-535)
We have seen (§ 5-4) that Q is equal to the product of m and twice the area swept out by the radius vector in the unit time. There- fore Q = 2m— (5-536)
T
, (5-54)
54 THEORETICAL PHYSICS [Ch. Ill
if T is the period of a complete revolution. From (5*535) and (5*536) we get
Johannes Kepler (1571-1630) inferred from the observational data accumulated by Tycho Brahe (1546-1601) that the planets travel round the sun in elliptic orbits, the sun being in one focus of the ellipse, and that the radii vectores sweep out equal areas in equal times. These inferences, known as Kepler's first and second laws, were published in 1609.
Equation 5*54 expresses Kepler's third law, namely that the cubes of the major axes of different planetary orbits are propor- tional to the squares of the corresponding periods of revolution. This was published by him a few years later.
A rather important proposition connected with elliptic motion under the inverse square law of force is the following : — The energy of the particle m is completely determined by the length of the major axes of the ellipse. The energy equation (5*51) reduces to
2mr
or r2-^r-ii-=0 .... (5-55)
E 2mE ^ '
when the planetary body, electron or whatever it may be, is just
dT at one or other end of the major axis, since in this case — = 0.
dt
If the corresponding values of the radius vector are r^ and r^,
we have of course
ri +^2 = 2a;
but from (5*55) we have
B
and therefore
^ = 1 (5.551)
which proves the proposition.
We have so far supposed the attracting (or repelling) centre to be fixed in space, or at all events fixed relatively to the axes of co-ordinates. We have now to amend equation (5*51) so that it will apply more generally to two mutually attracting or repelling bodies, not subject to other forces. Let M and m be their masses and B and r their respective distances from their
§5-5] INTRODUCTION TO DYNAMICS 55
common centre of mass in which we suppose the origin to be placed. The energy equation now becomes
Since r and U are the distances of the masses m and M respect- ively from the centre of mass,
rm = EM,
m B or = __ = (7
M r
and therefore (5*56) becomes
2
Therefore
This equation would be identical with (5*51) if in the latter m
were replaced by m' = m(l + cr) and B hj B' = ; so
that the problem of the motion of the particle m is reduced to the one we have already solved. In particular we find for the energy of the system
2a 2aM-{-m
where a is the semi-major axis of the ellipse in which the particle
m is moving.
An instructive example of a central force is that represented
7? O by —-+ — ^ where B and C are constants. The potential energy
is obviously
if we adopt, as usual, the value zero for the arbitrary constant involved. The energy equation is now
Imi — ] + 4- h — 7. = E . . . (5*57)
^ \dtj 2mr^ ' ^ ' 2r2 ^ '
66 THEORETICAL PHYSICS [Ch. Ill
and differs from (5*5 1) only in the expression for the potential energy. The same method as that which was used in the case of the inverse square law leads to
instead of (5*5 14), and to
instead of (5-52).
TThCJ
We shall restrict our attention to the case where 1 + -— -
is positive and not zero. The general solution of (5*58) may then be expressed in the form
mB
u = — _/'!^^ ^. + R cos -! ^ -^ "^ ^
■?7 (5-59)
where R and rj are constants of integration. We may write this equation in the form
Q^ + mC
mB
l+.cosj^/l+^
6
(5-591)
where both e and ^ / 1 + — —
may, without loss of generality,
be taken to be positive. It will be seen that (5*591) resembles the equation of a conic. This resemblance is made more obvious still if we write it in the following way :
Q^ +mG
mB
(5-592)
V'
1 + £COS {(9 - 7f}
where
The essential difference between the orbit represented by (5-531) and that represented by (5-592) lies in the fact that whereas in the former t^ is a constant the corresponding quantity rj' in the latter is continuously increasing or decreasing during the motion,
according as a / 1
mC If we give the axes of co-
is less than or greater than unity, ordinates a suitably adjusted motion
§6]
INTRODUCTION TO DYNAMICS
57
of rotation about the Z axis we may keep y] constant and evidently the orbit referred to such moving axes will he a conic section. We may say therefore that the orbit is a conic section the major axis of which is in rotation, in the plane of the motion, in the same direction as that in which the particle is moving, or in
the opposite one, according as a / 1 + greater than unity.
mC
is less than or
§ 6. Generalized Co-ordinates
The n independent numerical data, q^, q^, q^, . . . q,i, which are necessary for the complete specification of the configuration of a dynamical system, are called its generalized co-ordinates, and the system is said to have n degrees of freedom. Such co- ordinates may be chosen in various ways ; for ex- ample in the case of a single particle they may be its rectangular or polar co- ordinates. A rigid body, free to move in any way, will require six co-ordin- ates ; but only three if a single point in the body is fixed. Associated with each ^ is a generalized
velocity -?, and a gene-
at
ralized momentum usually represented by the letter p. A rigid body, the only possible motion of which is a rotation about a fixed axis, has only one q, which may conveniently be the angle between a fixed plane of reference, OA in Fig. 6, and another plane, OB, fixed in the body, the intersection of the planes coinciding with 0, the axis of rotation, which in the figure is perpendicular to the plane of the paper. The kinetic energy of any particle of mass m in the body is
Fig. 6
mv^ = ^mr
\dt,
where r is the perpendicular distance of the particle from the axis, 0, and (/> is the angle between r and OA. If the body is rigid
d(j) _ dq
dt ~"di'
58 THEORETICAL PHYSICS [Ch. Ill
and the kinetic energy of the particle is
Since (-r-) ^^.s the same value for all particles in the body, the total kinetic energy will be
T = i{Imr')(^^y (6)
The quantity Zmr^ is called the moment of inertia of the body with respect to the axis in question, and it occupies in formula (6) the same position as that of the mass in the ex- pression ^mv^ for the kinetic energy of a particle.
The generalized momentum corresponding to -— is defined by
dt
^ dt ^ '
This definition is in accord with the use of the term angular momentum in § 5-4 (see equation 5 '412). If the axis of rotation of a rigid body pass through the origin and if x, y and z are the co-ordinates of any particle of mass m, in the body, the components of the angular momentum of the body will be
^ / dz dy\ ^ / dx dz\
^» = M4-^J) • • • (^-^^^
see equations (5*41).
Equations (5*432) also will apply to a rigid body and may be expressed in the form
|
dt |
= 2{yF. - |
- ^F„), |
|
dt |
= 2(^F. - |
- xF,), |
|
dp, dt |
= ^^Fy ■ |
-yF,) |
. . . (6-02)
In these equations we may, just as in § 5-43, regard the forces as of external origin, since the forces of internal origin con- tribute nothing to Z{yF^ — zFy), etc. The right-hand members of (6 '02) are the components of the applied torque or couple.
§ 6-1] INTRODUCTION TO DYNAMICS 59
The parallelism between equations (6*02) and those for the motion of a particle should be noticed. The former state that Rate of Increase of Angular Momentum = Applied Couple, and the latter that
Rate of Increase of Momentum = Applied Force.
§ 6-1. Work and Eneegy
Let dq = [dq^, dqy, dq^) be a small rotation of the body- about an axis through the origin. We may represent dq by a straight line from the origin, of length dq and drawn in the direction in which an ordinary screw would travel with such a rotation. If dl = [dx, dy, dz) is the consequent displacement of a particle, the distance of which from the origin is r = (x, y, z). We have, numerically,
dl = r sin 6 dq,
where 6 is the angle between the directions of dq and r ; and when we study the directions we find
dl = [dq, r] (6-1)
The work done during the displacement dl is equal to the scalar product of F, the force acting on the particle, and dl the dis- placement. Therefore the work is
(dl F) = ([dq r]F).
Reference to equation (2' 14) will show that this is equal to
([r F] dq).
In a rigid body dq is the same for all particles, therefore the work done by all the forces acting on the body will be given by
(dqr[rF]), and the rate at which work is done will be equal to
This must be equal to the rate of increase of the kinetic energy of the body, so that we have
In the summation we need only take account of forces of ex- ternal origin since the contribution to Z[r F] of the mutual forces exerted by the particles of the body on one another is zero. Once more there is a close parallelism with a correspond- ing result in particle djniamics. Equation (6*11) states that the rate of increase of the kinetic energy is equal to the scalar
60 THEORETICAL PHYSICS [Ch. Ill
product of angular velocity and applied couple. The rate of increase of the kinetic energy of a particle is equal to the scalar product of its velocity and the force acting on it.
§ 6-2. Moments and Products of Inertia
If Jo and I, represent respectively the moments of inertia of a rigid body with respect to any axis, 0, and an axis C, through
the centre of mass of the body Y and parallel to 0,
I, = Mh^+I, . (6-2)
M being the mass of the body and h the perpendicular dis- tance between the two axes. To prove this it is convenient to place the co-ordinate axes so that the axes 0 and C are in the XZ plane and perpen- dicular to the X axis (Fig. 6-2). Fig ^gT^ '^^^ contribution of any par-
ticle m to Jo is mr^ _ jjiji2 _|_ ^^2 _ 2mlis cos (/> or mr 2 = mh^ + ms^ + 2hm{x — ^^o),
where x is the X co-ordinate of the particle and Xq is the X co-ordinate of the centre of mass. Summing over all the particles in the body we get
Jo = Mh^ + Jc + 2h{Zmx - Mxo},
and, by the definition of centre of mass,
31 Xq = Emx,
therefore equation (6*2) follows.
The radius of gyration of a body with respect to any axis is defined to be the positive square root of the quotient of the moment of inertia of the body with respect to that axis by the mass of the body :
h
M , so that
I = 3Ik^ (6-21)
Unless the contrary is expressly stated, or clearly implied by the context, we shall associate the radius of gyration with axes through the centre of mass of the body.
Let a, p and y be the direction cosines of an axis through
V 31
§6-2] INTRODUCTION TO DYNAMICS 61
the centre of mass, which we shall suppose is at the origin of rectangular co-ordinates. The contribution of any particle of mass m to the moment of inertia, /^^^, with respect to the axis (a, p, y) is mr^ sin^ 6, where r = {x, y, z) specifies the position of the particle and 6 is the angle between the directions of r and of {a, ^, y). Since r^ sin^ 0 is the square of the vector product of the vector r and the unit vector (a, ^, y),
mv^ sin^ d = m{(yy — z^Y + (s^a — xyY + {^^ — ya)^}-
Therefore
•^aiSv = a^Zm{y^ + z^) + P^Zm{x^ + z^) + y'^Zm{x'^ -\- y^) — 2pyZmyz — 2ayZmxz — 2apZmxy,
J,^^ = Aa^ + Bp + Cy^ + 2Dpy + 2Eay + 2Fa^ . (6-22)
The coefficients
A = Imiy^ + z^),
B = Em(x^ + z^)
and C = Em{x^ + y^)
are, clearly, the moments of inertia of the body with respect to the X, Y and Z axes respectively. The remaining coefficients
— D= Zmyz,
— E — Emxz and — F = Emxy
are known as products of inertia. Consider the surface ^|2 4_ Bri^ + CC2 + 2J[)^C + 2^^f + ^F^ri =M . . (6-23) where M is the mass of the body and i, tj and C are X, Y and Z co-ordinates. Let q be the length of a radius vector from the origin to a point (I^^C) on the surface, so that
^2 _ |2 _|_ ^2 _(_ ^2,
We have for a radius vector in the direction (a^y)
a = ^/q, P = r]/Q, y = C/q,
and therefore, dividing both sides of (6*23) by q^,
M Aa^ + 5^2 ^ Cy^ ^ 2D^y + 2Eay + 2FaP = — ,
so that
Q This means that the length of any radius vector of the surface
62 THEORETICAL PHYSICS [Ch. Ill
(6*23 ) is equal to the reciprocal of the radius of gyration of the body with respect to an axis in that direction.
The equation (6 '23) must be that of an ellipsoid, since in any direction whatever q wiU have a finite positive value. It is caUed the momental ellipsoid or ellipsoid of inertia. A suitable rotation of the axes about the origin will reduce the equation to
AP +Brj^ -\-GC^ =M . . . (6-235)
where A, B and G have not necessarily the same values as in
(6-23). A, B and G (6-235) are called the principal moments
of inertia of the body and the corresponding X, Y and Z
directions are called the principal axes of inertia.
M M In equation (6-235) let us replace |, y} and C by -—^', -—r\'
A B
M and -^C' respectively, so that we get the equation
A '^ B '^ G M
t'2 ,/2 ^'2
|^ + f. + F-.= l (6-24)
/t/j A/2 A/3
in which hi, kz and k^ are the radii of gjrration about the prin- cipal axes of inertia. The surface (6-24) is called the ellipsoid of gyration. Any point (^', rj', C) on it corresponds to a point (I, 7], f ) on the momental ellipsoid, where
^ A^'
M
n-^n' ' (6-241)
^ G^
The length of the perpendicular, P, from the origin to the tangent plane at (|' t]' ^') is, by a well-known rule,
J_ M
V
A^ B^ (72
or P == , ^ - -, by . . (6-241)
Vr+^2_|_^2| Q
and it is easily seen that its direction cosines are the same as those of the radius vector q from the origin to the point (i r] C)
§6-4] INTRODUCTION TO DYNAMICS 63
on the momental ellipsoid. It foUows that the length of the perpendicular from the origin to a tangent plane of the ellipsoid of gyration is equal to the radius of gyration of the body about an axis coincident with the perpendicular.
§ 6-3. The Momental Tensor
Equation (6*22) may be written in the form :
a{Aa +F^ + Ey} + P{Fa + BS -f Dy}
+ y{Ea-{-D^ + Cy}^I^^^ . . . : (6-3)
The right-hand member of (6*3) is an invariant, since it is the moment of inertia of the body with respect to a specified axis in it and must therefore be independent of the system of reference chosen. This suggests that the set of quantities
A F E F B D E D C (6-301)
are the components of a tensor of the second rank. It can easily be proved that this is the case (see § 2-3). It is called the momental tensor and is more appropriately represented in the following way :
V V i. (6-302)
If we make a corresponding change of notation for the direction cosines {a^y), equation (6*3), becomes :
•*■ a ^x \^x ^xx i ^y ''xy ~r ^z '^xz J
+ tty {a^ lyy, "J" tty I yy "f tt^ '^1/3 /
+ «.K hx + «y hy + «. %z} • • (6-303) The moment of inertia I^ is therefore to be regarded as the scalar product of two vectors, namely the vector (a^ a^ a^) and the vector, the components of which are represented by the ex- pressions in brackets in (6*303).
§ 6-4. Kinetic Energy of a Rigid Body
If the motion of the body is a rotation about an axis a, the
angular velocity being to = -^, its kinetic energy is T = J/a<^^
and since co^ = hsa^, o)y = isiay, co^ = coa^, we obtain from (6*303) T = i(o^{oj^ i^^ + ojy i^y + CO, *^J
+ i^y{(^x ijx + ^y Ky + ^z \z}
+ i^zi^x hx + ^y hy + ^\ iz} ' ' ' i^'^)
64 THEORETICAL PHYSICS [Ch. Ill
If the axes of co-ordinates coincide with the principal axes of inertia, this becomes
T = H^^x' ire + <^-^/ V + ^z" hz) ;
or, if we revert to the notation of (6*235 ) for the principal moments of inertia,
T = |{^co,2+^co,2 + (7a),2} . . . (6-41)
If in (6*4) we represent the angular velocity co by -~, we have
where p^ = ij^^ + ij^ + i,§,
^^ '""dt ^ ''dt ^ "'dt' dq^ . dq, dq,
''''di ^ ^''dt ^ ''W
Pz =
co^y - co,^x,
are the components of the angular momentum. In fact the X component of the angular momentum is
and -^ = cojjc — cojz,
dt
dz
di (see equation (6*1) for example) ; therefore
p^ = Zm{co^^ — cOyXy — co^xz + co^z^}
= Kxf^x + hy(^y + hz(^z'
It will be observed that
_dT _dT _dT
^qx Gqy dq, where q^ = co^ = _^, etc (6-42)
Any motion of a rigid body can be regarded as a motion of translation, in which all the particles of the body receive equal and parallel displacements, on which is superposed a rotation about a suitably chosen axis. Let r be the distance of any particle of mass m from a point, P, on the axis of rotation. A rotation dq will give it a displacement [dq r]. The total displacement of the particle will be the vector sum of [dq r]
§6-5] INTRODUCTION TO DYNAMICS 65
and the displacement of P. Let the co-ordinates of the particle be X, y and z and those of P be Xq, y^ and z^ ; then
T = {x -x^, y -y^, z — z^), and
dx = dxo + {dqy{z — ^o) — dq,{y — 2/0)} • (6-43) Therefore
% = ^ox + '^ix • • . • (6-435) Where v is the velocity of the particle, Vq the velocity of P and
^,, = J"(z-2„)-^%-2/„) . . (6-436)
The kinetic energy of the particle is
= imvo^ + imvi2 -f m{vo^Vi^ + VoyV^y + Vo^v^,), and the kinetic energy of the body,
T ^T, + T, + Zm{vo,v,^ + v^yV.y + ^^o^^ij . (6-44) In this equation
To = iMv,^ T, == ilco^
M is the mass of the body and / is its moment of inertia with respect to the axis of rotation.
If the point P is the centre of mass of the body,
since, as reference to (6*436) will show, it consists of terms, each of which contains one or other of the factors
Em[z — Zq), Em{y — y^), etc.,
all of which vanish if (a; 0 2/0 ^0) is the centre of mass. We thus arrive at the important result,
T = lilfvo^ + i/to2 .... (6-45) where Vq is the velocity of the centre of mass and co is the angular velocity relative to the centre of mass.
§ 6-5. The Pendulum
The pendulum is usually a rigid body mounted so that it can turn freely about a fixed horizontal axis, O, (Fig. 6-5), which we may suppose to be the Z axis of rectangular co- ordinates. The position of the pendulum is determined by the vector
q ^ {(ix cLv qz),
where q^ = qy = 0 and g^ = g is the angle between the plane XZ and the pla.ne, OC, containing the centre of mass and the
THEORETICAL PHYSICS
[Ch. Ill
axis of rotation. The positive direction of q is indicated in the figure by an arrow. The equations (6'02), when apphed to this case reduce to
or
yF.)
if the Y axis is directed vertically downwards, since the im- pressed forces on the body are due to gravity only. The force Fy on any particle is equal to mg, therefore
i^ = gEmx = Mgx^,
or
-/
Mgh sin Q,
Fig. 6-5
where 0 is the angle between OC and the vertical and h is the distance of the centre of mass from the axis of rota- tion. It is convenient to
write the equation in the form :
d^d , Mgh . , _
do
If we multiply by — and integrate, we get at
*(
dOy di)
Mgh
cosd = K
(6-5)
(6-501)
where ^ is a constant of integration. If K exceeds
Mgh
the
kinetic energy, |7( — ) can never sink to zero, and the body
win keep on rotating in the same sense round the axis with a periodically varying angular velocity. The case of interest to
Mgh
us is that in which K is less than
There wiU then be a dd
value ^0 of ^ between 0 and n for which -— is zero and
dt
Mgh
cos do = K,
^6-5] INTRODUCTION TO DYNAMICS
and consequently
, /de\ 2 3Igh , ^
(1)
where
67
'de\^ Mgh , . „ . 2 ,
_0 _6o
' ~ 2' '^ ~ 2"
The time required by the pendulum to travel from the position 0 = 0 to an extreme position 6 = :j- do or from 6 = ^ 0o to 0 =0 is
{dt^ IJL_ f
J V Mqh J
de
Mgh J V sin^ £o — sin^ e | and therefore the complete period of oscillation is
Mgh J Vsin2 £„ - sin^ e I
T = 4
(6-51
To evaluate the integral we introduce a new variable, </>, defined
by sin s = k sin 0,
where /c = sin £o- On substituting in (6*51 ) we get
7r/2
^~ ^ ~Mqh J Vl -/c^sir ^^ ' • • ^^'^^^)
sin ^(/)
The elliptic integral in (6*51 1) is now expanded by means of the binomial theorem, thus,
7r/2 ff/2
+ 2X6'^ ™
^+ . . .}
This can be integrated term by term, by using the well-known reduction formula
7r/2
r/2
[ sin2» <j>d<t>= — ?• [ sin2^-2^ dcl>.
0 0
We get, finally,
68 THEORETICAL PHYSICS [Ch. Ill
When the amplitude, ^o, is small, i.e. when k is small, the period,
is independent of the amplitude. This result might have been reached much more shortly by replacing sin 0 in (6*5) by 6 when 6 is small. The equation then becomes
cm . Mgh. _ ^ W^'^ I '
or, if we write co^ for the positive quantity
^, + (^^6=0 (6-522)
The general solution of this equation can be put in the form
l9 = ^ cos (co^ - ^) . . . . (6-523)
where A and ^ are arbitrary constants. Since d will repeat its values every time cot — cj) increases by 27t ; we must have
{co{t + r) - ^} - {ojt - cf>) = 27Z
cor = 271,
or T = — (6-524)
CO
This is identical with (6*521) when co is replaced by / — ~- •
The type of motion defined by (6*522) is called simple harmonic motion. It has the important property that the period is independent of the amplitude.
By making use of (6*2) we may give to (6*521) the form
V-
2. ■^' + ''
In the ideal simple pendulum, ^ = 0, A is the distance, usually represented by I, from the point of support to the bob, and therefore, for small oscillations
To = 271 J -
If Ti represents the still better approximation obtained by ignoring quantities of the order of ac* and higher powers of k, we have from (6*52)
=r.(i + y,
§6-5]
or since <c = sm
INTRODUCTION TO DYNAMICS
«y
= T„(l+isin2|),
T,=To(l+j|-) . .
since the difference between the squares of sin
(6-526)
- and ^ is of the
2 2
order of /<*•
Cycloidal Pendulum. We have seen that the period of the type of pendulum we have been studying is a function of the amplitude. It was shown by Huygens {Horologium Oscil- latorium) that the motion of a particle, constrained to travel along a certain cycloid, is strictly isochronous, i.e. the period is inde- pendent of the amplitude. The equations of motion of a particle, P, constrained to travel along a curve in a vertical plane, the XY plane in Fig. 6-51, are
^-7^ = Qx.
m
dp
Y
mg + a, (6-53)
Fig. 6-51
where Q^ and Qy are the components of the constraining force. If s is the distance travelled by the particle along the curve, measured from some arbitrarily chosen point, 0', the vector ds = {dx, dy) is perpendicular to the vector 0- Therefore
(&x + Qydy = 0
(6-531
Multiply the equations (6*53) by -— and ~ respectively and add,
dt cit
dx d^x , dy d^y dy , ^dx , ^dy
dt dt'
dt dt^
dt
'dt
dt'
or
m d ( /dx\^ , /dy\^] dy
11), and tb
m d ( /ds\ JJtlxdi)
by equation (6*531), and therefore m d ( /ds^ 2
mg cos £
ds dt'
70 THEORETICAL PHYSICS [Ch. Ill
ds if £ is the angle between dy and ds. Dividing through by m—
dt
we have
— 2 = 9^ cos £ (6-54)
The motion of the particle will be simple harmonic (see the definition 6*522), and its period consequently independent of the amplitude, if
cos 8 = — as (6-541)
where a is any positive constant. Equation (6*54) then becomes
^+"^^ = ^' and the period of the motion is seen to be
If I be the length of the simple pendulum, the small oscillations of which have the same period,
1
and (6*541) may be written as
cos £ = - ? (6-542)
V
This is the equation of the required curve. On differentiating it we get
ds = I sin £ ds
|
and therefore |
dx = I sin^ £ ds, |
|
dy = I sin s cos s ds. |
|
|
Consequently |
dx =-(1 - cos d)dd, 4 |
|
dy =— sin 0 dd ; 4 |
|
|
where 0 = 2e. |
On integrating we have |
|
X =l{d - sin 6) +A |
|
|
y =— cos 0 -{- B . 4 |
. . (6-543)
Let 0', from which s is measured, coincide with the origin. 0, so that X = 0 when y = 0 ; and suppose that the particle,
§6-5] INTRODUCTION TO DYNAMICS 71
P, is moving vertically downwards when in this position, i.e. £ = 0 = 0 when x = 0 and y = 0. For this position of the particle, therefore, equations (6*543) become
and on substituting in (6*543) we have the familiar equations of the cycloid
X =R{d - sin (9),
y = B(l - COB 6) (6-55)
in which R has been written for — .
4
BIBLIOGRAPHY
Galileo Galilei : Discorsi e dimostrazioni raatematiche, 1638.
Galileo's dialogLies have been translated by Henry Crew and Alfonso de Salvio. (The Macmillan Co., 1914.) Newton : Philosophise natnralis principia mathematica, 1687. Ernst Mach : Die Mechanik in ihrer Entwickelung, 1883. Huygens : Horologium oscillatorium, 1673.
CHAPTER IV
DYNAMICS OF A RIGID BODY FIXED AT ONE POINT
§ 7. Euler's Dynamical Equations
LET P be any vector and P^, Py and P^ its components referred to rectangular axes of co-ordinates. Let PJ, Py and P/ be the components of the same vector referred to a second set of rectangular co-ordinates, the origin, 0', of which coincides with 0, the origin of the first system. Therefore, by (2-22)
Px ^^ ^iiPx ~i" (^2iPy ~r CisiPz
and ^- = |{a,,P; + a,iP/ + a„P/} . . (7)
We shall suppose the first set of co-ordinates to be fixed and the second set to be in motion about their common origin. The cosines an, a 21, etc., are then variable and (7) becomes dP. dPJ , dPJ . dPJ
+ a,^—^ + a
3r
dt dt dt dt
p ,aaii p ,0021 I p /dosi //v.ooi "i
"^ ^ dt '^ ' dt '^ ' dt ^^ ^
Now a 11, a 21 and a 31 are the co-ordinates in X\ Y\ Z' of a point
on the X axis at the unit distance from the origin, and therefore
— yi^, —7^ and — -^ are the components of the velocity of this
point relatively to the moving axes. Therefore if co^', my and m^ are the components of the angular velocity of the fixed co-ordinates relatively to the moving co-ordinates,
dt da
= co^ asi — CO, ttai = co.aai — oy^a,
dt da^i
21 = co/an — ctyJa^i
o^x "21 — ojy «ii = cOyaii — oy^fL^ii
dt "''"'' '
72
§7] DYNAMICS OF A RIGID BODY 73
where co^ = — co^;'? ^y ^^ ~ ^y ^^^ <^2 — "~ ^z\ so that m^, (Oy and co^ are the components of the angular velocity of the moving axes X', X' , Z' relatively to the fixed axes X, X, Z.
If we now substitute these expressions for --^, etc., in equations (7*001) we get,
dP, dPj ^ dp; , dp; -dt ^ ""''-df + ""''-df + ""''-W
+ P;(a>,aii - co.a.i) . . (7-002) At an instant when the fixed and moving axes are coincident,
«ii = 1, a^i = aai = 0 and equation (7*002 ) becomes
dt dt
to which we may add
i = :^ + p;co, - P>,.
dP dP '
^ = ^ + P' ny, - PJoy^ , . . (7-01 )
dt dt "" ' ' "^ ^ '
dP, ^ dP,' dt dt
It is very easy to be misled by these equations, and we shall therefore inquire carefuUy about their significance before applying them. In arriving at the transformation (2-22) we represented the vector concerned (in the present case P) by a straight line drawn from the origin in the direction of the vector, and having a length numerically equal to it. Therefore P^, Py and P^ are the co-ordinates of the end point of the line. Equations (7*01) apply at the instant when the two co-ordinate systems coincide. Hence P„ = P/, Py = P/, P, = P/. Suppose now that P is the angular velocity of a rigid body with one point fixed in the common origin of the co-ordinate systems. Clearly the components of the angular velocity of the body have the same values in both sj^stems of co-ordinates when they happen to coincide. It is important to note this and so avoid the error of confusing the angular velocity referred to the moving axes with the angular velocity relative to the moving axes. In fact, if the moving axes were fixed in the rigid body, its angular velocity would be (CO3,, My, coj in both systems of co-ordinates ; but obviously zero relative to the moving axes ; and we note too that the rate of change of co is the same whether referred to the fixed or the moving axes, as is immediately evident on substituting CO for P in (7-01).
74 THEORETICAL PHYSICS [Ch. IV
Let us now suppose the moving axes to be fixed in a rigid body and to coincide with its principal axes of inertia through the fixed point of the body (the common origin of both systems of co-ordinates) and let us further suppose P = {P^, Py, PJ to
be the an2;ular momentum of the body. Then ( ~, ~—^, -— ? ) ^ -^ \ dt dt dt J
becomes the torque or couple applied to the body. In what
follows we shall denote this by (L, M, N), At an instant when
the axes are coincident
P, = P,; = Am,,
p, = p; = Ceo,,
where A, B and C are the principal moments of inertia of the body. Clearly
dPJ _ jdoy, dt dt '
dPy' ^ -^dwy dt dt '
dPJ ^ (7^ dt dt '
On making these substitutions in equations (7*01 ) we obtain
i = 4^- + (C - B)o>,oy^,
N = Cf^ + (B - A)c^co, . . . (7-02)
When the applied couple vanishes these equations become ,dco^ dt
^-^ = iB- (^H^.
b"^^ = {G - A)co,co,,
C^ = {A - B)m,cOy . . . (7-021)
The equations (7*02) and (7*021) are the well known dynamical equations of Euler.
On multiplying (7-02) by ca,, cOy, and w^ respectively and adding, we get
§7] DYNAMICS OF A RIGID BODY 75
which states that the rate of increase of the kinetic energy of the body is equal to the rate at which the applied couple does work, a result we expect on other grounds (equation 6*11).
When the couple applied to the body is zero, i.e. when L = M = N = 0, we find, by multiplying equations (7*02 1) by Aco^, BcOy and Cco^ respectively and adding
or A^co^^ + B^co/ + C2ca,2 = Q^ . . . (7-04)
where D^ is a> constant. This equation is also to be anticipated on other grounds, since it expresses the constancy of the angular momentum (§6).
A particular solution of equations (7*021 ) is
(^x — <^v — ^ '^ <^z — <^05 ^ constant. This represents a rotation with constant angular velocity about a principal axis of inertia. Suppose the body to be rotating in this way and then slightly disturbed, so that it acquires very small angular velocities co^ and cOy about the other principal axes of inertia. How will it behave if it is now left to itself ? Since CO3. and cOy are small (i.e. by comparison with coo), we shall ignore the product co^(Oy. Euler's equations now become
^W "^ ^^ ~ ^^^'''^' = 0 . . . . (7-05)
B^ + (A- C)co,co, = 0.
Differentiating the former of these with respect to the time, and eliminating — -^, we obtain
W + (£z1^^.lAW..,.0., (7.051)
By differentiating the second of the equations (7*05 ) in a similar way we obtain
^. + (^-^^^^Wco„ = 0 . (7-052)
The constant
iC-B){C-A)
AB
in both of these equations is positive if the moment of inertia G is either greater than A and B or smaller than A and B. In such a case
0)^ = B cos {at — (f))
o)y=S COS {at -y)) . . . (7-053)
76 THEOHETICAL PHYSICS [Ch. IV
where R and >S' are small real constants, (/> and xp are constants, and
coo . . (7-054)
J
{G~A){G-B)
AB
We see therefore that the motion of rotation is stable since cOg, and ojy never exceed in absolute value the small constants R and 8.
It should be noticed that R, S, ^ and y) are not all independent. The reason for this is that equations (7'051) and (7'052), in the solutions of which they occur, are more general than the equa- tions (7*05) with which we are really concerned, since they are obtained from the latter by differentiation. If we abbre- viate by writing
p = at — cj), q = at — ip,
and substitute the solutions (7*053 ) in equations (7*05), we get sin p (G — B)Soyo
and therefore
|
cos |
Q |
aAR ' |
|
|
sin |
p |
_(^ |
- G)Rm, |
|
cos |
aBS |
||
|
sin |
p |
sin q |
(7-055)
cos p COS q in consequence of (7-054).
It follows that p and q differ by an odd multiple of — and
the solutions (7-053) may consequently be put in the form
0)^ = R cos (at — cf)),
C0y==8 sin [at -cf>) ... (7-056)
the first of the equations (7-055) now becomes
(G - B)Scoo
1
ARa
whence we get
^ = ^70^ • • • ^'-'"'^
If we represent the angular velocity o> by a straight line drawn from the origin, equal in length to to, and in such a direc- tion that the co-ordinates of its end points are co^, cOy and co^ (= (Oq) respectively, we see that it describes a small cone in the body. The end point travels along the small ellipse with semi-axes R and S.
§ 7] DYNAMICS OF A EIGID BODY 77
The form of equations (7*021) suggests that their general solution can be expressed in terms of elliptic functions. Con- sider the integral
e
} dd
^^ J Vl-y^2 singer ^'<^'
0
which belongs to the class of integrals called elliptic integrals. The upper limit, 0, is termed the amplitude of | and may be denoted by am §• Therefore
sin 6 = sin am |, or, in the usual notation
sin 6 = sn ^. Similarly cos 6 = cos am i = en i.
The function Vl — k^ sin^ 6\ is usually called A^?
/)^d = /\ am ^ = dn ^. The three functions, sn^, cn^ and dni are called elliptic func- tions. The differential quotient of sni with respect to | is
d sn i _d sin 6 dO ~~dl dd~'di'
i^^Go^Q.Vl -yb^sin^^l, d I
dsn ^ f. J f.
or — -— - = en i dn |.
di
Similarly
d en ^ f. T ^
—nr- = — sn^dn^, d^
i^ = -k^snicn^ . . . . (7-06) a f
These equations suggest, as a solution of (7*021 ), <^x = coi sn {at — (f)), cOy = (O2 c^ {Git — ^)j CO, = (Oodn {at — (j)) .... (7'07)
where coi, CO2, coq, a and ^ are constants, which, as we shall see, are not all independent. Substituting in (7*021 ), we find
B -G
acoi = — cogCOo
— aft) 2 = — Fi — coiCOq
— ak^coo = — -^ — ft)ift>2 . . . (7*071) G
|
A |
u |
|
|
G |
— |
A |
|
B |
||
|
A |
— |
B |
78 THEORETICAL PHYSICS [Ch. IV
Therefore
coi^ {G - B)B
m^ {C-A)A'
{A-B)Am,'^ (C -B)C coo'
(7-072)
2 _ (^ -B)(G-A)^_^ ,
and a^ = -^: — — coq
Of the six constants, coi, coa, oy^, k, a and ^ therefore, three can be expressed in terms of the remaining three. These latter may be chosen arbitrarily and the solution (7'07) is therefore the general one. Let us select a>i, coq and cf) as the arbitrary constants and consider the case where coi and co^ are very small compared with (Oq. The parameter k^ will be a small quantity of the second order, by the second equation (7'072). We shall therefore ignore it. We thus get
in the equations defining the elliptic functions. Therefore
sn i = sin 6 = sin |, en ^ = cos 6 = cos |, dn i = I,
and the solution (7-07) reduces, as of course it should, to that already found for this special case (equations 7*056 and 7*057).
§ 7-1. Geometuical Exposition
We have in (7*07) the solution of the problem of the motion of a rigid body, one point in which is fixed, for the special case where the forces acting on the body have no resultant moment about the fixed point. A very instructive picture of the motion is provided by the geometrical method of Poinsot (Theorie nouvelle de la rotation des corps, 1851). The results we have aheady obtained indicate that the instantaneous axis of rotation wanders about in the rigid body and therefore sweeps out in it a cone {s, Fig. 7-1), having its apex at the fixed point, 0. The positions of this axis in the body at successive instants of time are represented by Oa, 06, Oc, Od, Oe, etc. The lengths of these lines may conveniently be made equal, or proportional, to the corresponding values of co at these instants. During the time required by the axis of rotation to travel from Oa to 06 the point 6 will travel in space to some point ^. That is to say, the line 06 in the body will occupy the position 0/9 at the instant when it coincides with the axis of rotation. In a succeeding interval the axis of rotation will have reached Oc, (in the body) which
§7-1]
DYNAMICS OF A RIGID BODY
79
wiU now have a position Oy in space, and so forth. The lines, Oa, 0^, Oy, 0^, Oe, etc., sweep out a cone, a, which is fixed in space. The motion of the body is consequently such as would result if a certain cone, s, fixed rigidly in the body, were to roll, with an appropriate angular velocity, on another cone, a, fixed in space. The cone, s, wiU cut the momental ellipsoid (which may likewise be described as fixed in the body, or rigidly attached to it) in a closed curve, as will be shown. This curve Poinsot called the polhode {noXoQ, axis ; odog, path). Its equations can be found in the following way : Using {x, y, z) in place of (^, y}, C) in the equa- tion (6*235), of the momental ellipsoid, we have for the com- ponents of the angular velocity, cj,
Fig. 7-1
CO,
CO
■X, COy
CO
-y, (^z
CO
—z
(7-1
Q - Q Q
p meaning, as in § 6*2, the radius vector from 0 to {x, y, z). The perpendicular, p, from 0 to the tangent plane at {x, y, z) is
p = Q cos d, if 6 is the angle between p and p. Therefore p is the scalar product, (p N), of p and a unit vector N in the direction p. Consequently
p = xa ^ yP -\- zy, if a, /5 and y are the components of N, or the direction cosines of p. The equation of the tangent plane at {x, y, z) is Ax^ + Byt] + CzC = M, if (I, rj, C) is any point on it. Therefore
Ax
VA^x^ +B^y^ + GV\
P
y =
p ^
By
VA^x^ +B^y^ + C^z^l Cz
VA^x^ + B^y^ + CV\
M
(7.11)
80 THEORETICAL PHYSICS [Ch. IV
But by combining (7*04) and (7*1 ), we find that
A^x^ +B^y^ -\-C^z^=^^ . . . (7-12)
Therefore i^ = ^ (7-121)
Similarly, by combining (6*41) and (7*1) we obtain
Ax'' + By^ + Cz^ = 2T^ . . . (7-13)
or M=2T^ . . .(7-131)
It follows that — is a constant, namely
and consequently
CO
P
(7-132)
^^V2TM\ . . _ .(7.133)
a
It is therefore constant and its direction cosines (7-11) are the same as those of the angular momentum £1. Consequently, it is invariable in length and direction, and the tangent plane remains fixed in space during the motion of the body. The last of the equations (7-11) gives us
AH^ -{- B^y^ -^ CV = ^ . . . (7-14)
This equation holds for any point (x, y, z) where the axis of rota- tion cuts the ellipsoid of inertia and it, together with the equation of the ellipsoid,
Ax^ +By^ -\-Cz^=M . . . . (7-15)
determines the polhode.
If we multiply (7-14) by p^ and (7-15) by M and subtract, we get
(p2^2 _ MA)x^ + tp2^2 _ MB)y^ + (p^C^ - MC)z^ = 0 (7-16)
which is the equation of the polhode cone s.
The curve traced out on the fixed tangent plane by the instantaneous axis of rotation was called by Poinsot the her- polhode (from squelv, to crawl, like a serpent). The corre- sponding herpolhode cone is the cone a, fixed in space, on which the polhode cone rolls. We have now a very clear picture of the motion, especially if we remember (7-132) that the angular velocity about the instantaneous axis is proportional to
§ 7-1] DYNAMICS OF A RIGID BODY 81
p, the radius vector of the momental ellipsoid which coincides with the axis. The cone, s, fixed relatively to the ellipsoid rolls on the cone, a, in such a way that the ellipsoid is in contact with a fixed plane, the velocity of rotation at any instant being proportional to the distance, q, from the fixed point, 0, to the point of contact with the fixed plane.
The semi-axes of the ellipsoid of inertia are
Suppose,
|
A>B>G, |
|
|
then |
f-'-^i |
|
In one extreme case |
|
|
--I |
(7-17)
and the equation of the polhode cone s (7' 16) becomes
(B^ - AB)y^ + (C2 - AC)z'' = 0.
Since both terms on the left of this equation have the same sign,
the only real points on it are the points y — z = 0, and the cone
reduces to a straight line, or, strictly speaking, to two imaginary
planes intersecting in a real line, the X axis. There is a similar
M state of affairs if ^^ j^^s the other extreme value — . If however
G
2 if
^' = F'
the equation of the cone becomes
(^2 _ AB)x'' + ((72 - 05)22 _ 0^
In this equation A'^ — AB is positive and (7^ — CB is negative. It therefore represents two real planes intersecting in the Y axis. Instead of combining equations (7*14) and (7*15) to get the equation of a cone, let us eliminate x'^. We thus obtain
[B'^ - AB)y'' -\- {G'' - AG)z'' = ^ - AM . (7-18)
Reference to (7*17) will show that the right-hand member of this equation is negative or, in the extreme case, zero, and since this is true likewise of the coefficients of ^/^ and s^, we conclude that the projections of the polhodes on the YZ plane are ellipses.
%i^^^
82 THEORETICAL PHYSICS
The ratio of the semi-axes of any of the ellipses is
lB(A
\ C{A
B{A~B)\
[Ch. IV
(7-181
Similarly, we can show that the projections of the polhodes on the XY plane are the ellipses,
(^2 _ AC)x^ + (52 - BG)y^ = ^-. MG . (7-182) and ratio of the semi-axes being in this case
V
A(G -A)
(7-183)
B{G -B)
This result should be compared with (7-057).
The projections on the XZ plane are the hyperbolas
[A^ - AB)x^ + (C2 - GB)z^ = ^ -MB (7-184)
§ 7-2. Efler's Angular Co-ordinates
We shall continue to use a system of axes, X', Y', Z', fixed in the body, and coincident with its principal axes of inertia.
Let X, Y, Z, be another set of axes fixed in space, the Z axis being directed vertically upwards, and the two sets of axes having a common origin, 0, in the fixed point of the body. Let the angle
§7-2]
DYNAMICS OF A RIGID BODY
83
between Z and Z' be denoted by 0. The X'Y' plane intersects the XY plane in the line, OH, (Fig. 7-2). The angle between OH and OX is denoted by ^, and that between OX' and OH by ^. The positive directions are indicated in the figure by arrows. The position of the body, at any instant, is completely determined by the values of these three angles, called Euler's angles.
The Eulerian angles are illustrated by the method of mounting an ordinary gyroscope (Fig. 7-21). There is a fixed ring, ABC. Within this is a second ring ahc pivoted at A and B so that it can turn about the vertical axis, AB. The axis AB corresponds to OZ (Fig. 7-2). Within the ring, ahc, is stiU another ring, a^y, pivoted at a and 6, so that it can turn about the horizontal axis, ah. This axis corresponds to OH (Fig. 7-2). The gyroscopic wheel, itself, is pivoted at a and ^ in the innermost ring, so that it can spin about an axis, a/5, perpen- dicular to ah. The axis, a/?, corresponds to OZ'.
Let us now express the components, co^, ca,,
and oi.} of the
angular velocity of the body, in terms of -j-, j- and ^. It is
dt dt
dt
dcf>
clear that co^ and o^y do not depend on ^ and we must therefore
az
have
oy.
CO,
= ^ COS {ZY') + ? cos (HY').
az az
Obviously m^ is not identical with -^ since ^ is an angle measured
az
To get cOg we have to add to ~ the
az
from the moving line OH.
angular -j- multiplied by cos (ZZ'), therefore
^ Note that cox, (Oy, coz have the same meaning as in Euler's equations. They are the components of the angular velocity referred to axes X', Y', Z' fixed in the body.
84
THEORETICAL PHYSICS
[Ch. IV
The direction cosines in these equations are easily seen to have the values set out in the table :
|
X' |
Y' |
Z' |
|
|
z |
sin 6 sin ^ |
sin Q cos (ji |
cos 6 |
|
H |
cos (j) |
— sin 0 |
0 |
For example, cos [ZX') is X' co-ordinate of a point on Z the unit distance from 0. The distance from 0 of the projection of this point on the X'Y' plane is sin Q and the angle between this projection and OX' is obviously the complement of ^. Hence we get the projection on OX' by a further multiplication by sin (j).
We therefore arrive at the following relationships : —
dw . ^ . J . dd J 0)^ = -J- sm & sm ^ + — cos ^,
Clt U/t
dw . ^ , dd . , cOy = -r^ sm 0 cos 0 — —- sm A, " dt dt
dip ^ , d(h CO, = -^ cos (9 + -^. dt dt
(7.2)
§ 7-3. The Top and Gyroscope
We shall now apply Euler's equations to the problem of the symmetrical top (or gyroscope) supposing the peg of the top (or the fixed point in the gyroscope) to be fixed in the origin. If the Z' axis is the axis of symmetry of the top, and if the distance of the centre of mass from 0 is h, the couple exerted has always the direction OH, and is equal to mgh sin d, m being the mass of the top. We must substitute for L, M and N in Euler's equations the components of this couple along the directions Z', 7' and Z' . The table of cosines (§ 7-2) gives us
L = mgh sin d cos 0, M = —mgh sin d sin 0, iV^ -0.
§7-3] DYNAMICS OF A RIGID BODY 85
On substituting these values for L, M and N in Euler's equations, (7'02), we have
mgh sin 0 cos ^ = ^ -~ + (C — B)cOya}y,
Cut
— mgJi sin 6 sin ^ = B-yJ + (A — 0)G>/a„
0 = Cf^-^ + (B - A)co^m,. . (7-3)
If we replace co^, cOy and co^ in (7'3) by the Eulerian expres- sions (7*2) we obtain three differential equations the solution of which gives the character of the motion of the top. Instead of proceeding in this way it is simpler to make use of the energy equation, and obtain two further equations by equating the angular momenta about the Z and Z' axes to constants. This we are at liberty to do, since the applied couple is in the direction OH, that is to say, in a direction perpendicular to Z and to Z', so that its component in either of these directions is zero.
We obtain the energy equation by multiplying equations (7*3) by 0)^, (Oy, and cOg respectively and adding. In this way we get
— {^Aco^^ + iBcOy^ + iOco/} = mgh sin 6(co^ cos cf) — cOy sin ^). clt
If now we write A = B, on account of the symmetry, and sub- stitute for (o^ and ca^ their Eulerian values (7*2), we have
l(*4^»'»(S)'+(§)"]+*M =•"'*»■»
in which we have replaced co^^ by coq^, which is a constant by the third of the equations (7*3). Thus on integrating we arrive at the result
"»■»©" +©"="-¥-» • <""
where a is a constant of integration. This is the energy equation. The table of direction cosines (§ 7-2) gives for the angular momentum in the Z direction,
Aco^ sin 6 sin ^ + BcOy sin 6 cos ^ + Cco^ cos 6, or A sin d{(o^ sin cf) -{- cOy cos </>) + Gcoq cos d.
On replacing cOg. and cOy in the usual way by the expressions in (7*2), we get for the angular momentum about the Z axis,
A sin Ofsin 6 ~\ + Ocoo cos 0 = a constant,
or sin2 6>^ = a-^°cos6l . . . . (7-32)
dt A ^ '
where a is a constant of integration.
86 THEORETICAL PHYSICS [Ch. IV
For the third equation we have
Cm^ = a constant, or, by (7-2),
C\ cos 6 -^ -\- -^\ = a constant, I dt dt)
or finally
coseg + ^^=co„ .... (7-33) dt dt
The three equations, 7'31, 7*32 and 7*33 completely describe the behaviour of the ordinary top, when its peg is prevented from wandering about, or the motion of any rigid body with axial symmetry (gyroscope), when one point on the axis is fixed in
space. By eliminating -~ from (7*31) and (7'32) we arrive at
the equation
If Coy, J 2 /f^0x2 2mgh ^ ,^^^,
We can simplify this and the remaining equations by the follow- ing abbreviations :
^ = ?^, 6=^,/. =cos9 . . (7-345)
and consequently
-^^rt- ■ ■ ■ (^-^4^)
We have therefore, when we substitute in (7'34),
'^/*\' _ ^^ _ ^^)(i _ ^2) _ (^ _ 5^)2 . (7.35)
(i)=<°
an equation which may be expressed in the integrated form
t= f , ^/^ (7-351)
J A/(a - /S/i)(l - /i^) _ (a _ 6„)2
C
if c is the value of ju at the instant ^ = 0.
Equations (7'32) and (7'33) take the respective forms,
dip _a -hfi n'Z(y\
dt 1-/^2 ^ ^
#_^^_M^-6^ .... (7-37) dt I — fj,^
The integral (7'351) belongs to the class of elliptic integrals, and therefore /u, or cos 6, is an elliptic function of the time and
§ 7-3] DYNAMICS OF A RIGID BODY 87
consequently oscillates periodically between a fixed upper limit ^oj and a fixed lower limit ^i. Otherwise expressed, the angle, 0, between the axis of symmetry and the vertical, will change in a periodic fashion between a smallest value Oq and a greatest value d 1. This motion of the axis is called nutation. The motion
expressed by -~, that is to say, the motion of the axis, OH
CLZ
(Fig. 7-2), is called precession.
We can easily learn the general character of the motion from equations (7'35), (7*36) and (7'37) without making explicit
use of the properties of elliptic functions. If we denote (-r-)
hy f{/x), equation (7'35) becomes
f{f,) = {a-M{l-f,^)-(a-bf,)\
Since ^ is a positive constant,
/(_00) = - CO, /(+ OO) = + 00,
and further
/(-l) = -(a + 6)^ f(+l) = -(a-b)K
Therefore /(— 1) and/(+ 1) are necessarily negative (or zero),
and the general character of the function f(jbi) (=(-7-) )
is that illustrated by Fig. 7-3. Only positive values of f(jLi) and values of fi between — 1 and + 1 can have any significance in the motion of the top. The significant points in the diagrams (Fig. 7-3) are therefore those in the shaded areas of a, b and c. During the motion fi varies backwards and forwards between fixed upper and lower Limits, /liq and ^j respectively and associated with this is a corresponding variation of the angle, 0. At the
same time the precessional velocity, — will also vary periodically
with the same period as jti (see equation (7*36)). If we restrict our attention to the case where a and b are positive, we have the following possibilities : (1) if 6 is small enough, i.e. if the top
is not spinning fast (see (7*345)), -~ will remain positive (7'36) ;
cit
(2) if the top is spinning very fast (6 large enough) -^ may change
Coo
sign during the nutational motion between 61 and Oq. This will happen when fj, is equal to -. A special case, (3), is that in 7
88
THEORETICAL PHYSICS
[Ch. IV
a
which [Xq = -. This is the case when the top is set spinning
and released in such a way that initially -r^- = 0 and -^ = 0.
at at
filA
-I
(at*/
M
(a)
'\|_j/"-¥
AtA
f(fA
(0
+1
/"
Fig. 7-3
J"(/^)
The three cases considered are illustrated by {a), (b) and (c) respectively in Fig. 7-31, which exhibits the curve traced out by the centre of mass on the sphere with centre, 0, and radius h.
Fig. 7-31
The figure illustrates the possibilities in the case of the ordinary top, for which ju is always positive.
It is instructive to consider the case where the top is not spinning and where the angular momentum about the vertical
§7-3] DYNAMICS OF A RIGID BODY 89
axis, Z, is zero, so that coq and a are both zero, (7*345) and (7*32 ). We have then the equation
m = {^^' = (,o.-M{^-i^
or if we substitute cos 0 for ix,
As ^ = — ~- we may express the equation in the form ■./dd\^ . mgh ^ a
This equation is seen to be identical with (6*501), since A and / have the same meaning, and the difference in sign is merely due to the fact that 6 in the one equation is the supplement of 6 in the other. The motion is that of the pendu- lum. We find, just as in ^ Q'5, two sub-cases. If - exceeds
—K-, i.e. if a exceeds ^, — never vanishes and body rotates con- 21 dt
tinuously round a fixed horizontal axis, but with a periodically
varying velocity. This is the case illustrated by (c) in Fig. 7*3,
since ( y- ) = fii^) vanishes at the points /^ = + 1 and fi = — I
and nowhere else between these limits. On the other hand, if a<^ /5, /(^) again vanishes at + 1 and — 1, (7*35) since a and
b are both zero, but also at a point, ^ = -, between these limits.
P This corresponds to the ordinary pendulum motion and is illus- trated in (6), Fig. 7-3. The significant values of (.i extend from
— 1 (when the pendulum is vertical) to -.
P Another interesting special case is that for which the interval
/^o/^i within which /(//) is positive is contracted to a point, so that the curve for/(^) touches the [x axis as in Fig. 7-3 {d). Therefore
11 is constant during the motion and consequently so is -J-.
The axis of the top or gyroscope sweeps out a circular cone in space with a constant angular velocity. Let us consider the case where [x is zero and the axis of symmetry therefore horizontal.
We see (7*36) that -~ is equal to the constant a. It is an instruc- az
tive exercise to determine a by means of equations (7*01). In
90
THEORETICAL PHYSICS
[Ch. IV
these equations we must remember that [m^, cOy, a>^) represents the angular velocity of the moving axes and not necessarily that of the gyroscope. Suppose the X' directed vertically upwards, Fig. 7-32, and fixed, while the axis of symmetry of the top or gyroscope coincides with the Z' axis. We have therefore for
(ft)^, COy, coj of equations (7*01) co^ = a.
a
0,
CO.
CO, =0.
Y
Let PJ, Py and PJ be the components of angular momentum relative to these axes. Then P '
Fig. 7-32
. 0, P ' =0
PJ = Ceo,
where coq has the same meaning as before.
dP^ dt
We have further
0,
dP, dt
= — mc
since these quantities represent the rate of change of angular momentum with respect to the fixed axes, which are momentarily coincident with the moving ones. The equations (7*0 1) are then satisfied if
— mgh = — Ccjo^a,
mgh
or
Ceo,
C^o
We can of course arrive at this result in a much simpler way. Let OA (Fig. 7-33) represent the angular momentum Ccoq at any instant. The applied couple will produce in a short interval dt a change of momentum dO, at right angles to OA, as shown in the figure. The angle dyj swept out during dt will therefore be
or
dQ
Ccoq
dn
'di
Ccoq
= dip
dip dt'
§7 4] DYNAMICS OF A RIGID BODY 91
But in the present case -r- is constant and equal to mgh, therefore
dip _ _ mgh dt C(Oq
as we have already found.
§ 7-4. The Precession" of the Equinoxes
The earth behaves like a top. The attraction of the sun is exerted along a line which does not pass through the centre of mass of the earth except at the equinoxes. It thus gives rise to a couple tending to tilt the earth's axis about its centre of mass and make it more nearly vertical. The state of affairs is very similar to that we have just been studying. The centre of mass of the earth corresponds to the fixed point, 0, the peg of top. Consequently the earth's axis exhibits a motion of nutation and precession. The line in which the equator cuts the ecliptic corresponds to the OH in Fig. 7-2. The points where it cuts the celestial sphere are caUed the equinoctial points, from the circumstance that day and night are equal in length when the sun passes through them. In consequence of the precession the equinoctial points travel slowly round the heavens in the plane of the ecliptic in a retrograde direction, a whole revolution requiring a period of 25,800 years. Associated with this is a corresponding motion of the celestial poles which in the same period describe circles of 23° 27' in diameter round the poles of the ecliptic.
BIBLIOGRAPHY
EuLEE, : Mechanica, sive Motus Scientia analytice exposita, 1736.
PoiNSOT : Theorie nouvelle de la Rotation des Corps, 1851.
Gray : A Treatise on Physics. (Churchill.)
Webster : The Dynamics of Particles and of Rigid, Elastic and Fluid
Bodies, (Teubner.) Cbabtree : Spinning Tops and Gyroscopic Motion. (Longmans.)
1
CHAPTER V PRINCIPLES OF DYNAMICS
§ 8. Principle of Virtual Displacements
N order that a particle may be in equilibrium, the resultant of all the forces acting on it must necessarily be zero. If F ^ {F^, Fy, F^) be the resultant force,
F = F = F = 0
This condition may be stated in the following alternative way :
(F 61) = FJx + Fydy -}- F,dz = 0 . . (8)
where 81 = {dx, dy, dz) is an arbitrary small displacement of the particle, i.e. any small displacement we like to choose. For sup- pose we assign to dy and dz the value zero, and to dx a value different from zero. Equation (8) then becomes
F^dx = 0,
and hence F^ = 0. Similarly the statement (8) requires Fy and F^ to be zero.
Consider any number of particles, which we may distinguish by the subscripts 1, 2, 3, ...<§,... , and let the respective forces acting on them be Fj, Fg, Fg, . . . F^ . . . Further, imagine the particles to suffer the arbitrary small displacements, {dx^, dy^, dz^), {dx^, dy^, dz^), . . . (dx„ dy„ dz^), . . . Then the condition for the equilibrium of the system of particles is
^{FJx,+FJy, + FJz,)=0 . . (8-01)
the summation being extended over all the particles of the system. The arbitrary small displacement (dx, dy, dz) is called a virtual displacement and the statement (8) or (8*01) is called the principle of virtual displacements or the principle of virtual work.
The utility of the principle becomes evident when we apply it to cases where the particles are subject to constraints. As an illustration consider the case of a single particle so constrained that it cannot leave some given surface. There will in general be some force, F' = (FJ, Fy, F/), normal to the surface, and
92
§8] PRINCIPLES OF DYNAMICS 93
of such a magnitude that it prevents the particle from leaving it. Let us write equation (8) in the form
{F, + FJ)dx + {F, + F,')dy + {F, + F:)dz = 0,
where F = (F^, Fy, F^) represents the part of the force on the particle not due to the constraint. We shall call it the impressed force. Of course the principle of virtual displacements requires that
F, + F,' =F,-h Fy' =F,+F: =0;
but this is not the most important, nor the most interesting inference from the equation. If we subject the virtual dis- placement 51 = (dx, dy, dz) to the condition that it has to be along the surface, we have, since F' is normal to the surface,
(F'51) = FJdx + Fy'dy + F.'dz = 0 and consequently
FJx + Fydy + F,dz = 0 ... (8-02)
In this statement of the principle all reference to the force F' due to the constraint is eliminated ; but in applying it we have to remember that the virtual displacement is no longer arbitrary, and we cannot infer therefore that F^ = Fy = F^ = 0. Indeed, this would in general be untrue. Let the equation of the surface, to which any motion of the particle is confined, be
cl>{x,y,z)=0 (8-03)
The virtual displacement is therefore subject to the condition
|.. + |., + |.. = 0. . . (8.031)
Let us eliminate one of the components of 81, e.g., dx, with the help of (8*02). We can do this most conveniently by multi- plying (8*031) by a factor, A, so chosen that
i^. - A^ = 0 . . . . (8-032)
and subtracting the result from (8*02). This gives
The components dy and dz of the virtual displacement can be chosen arbitrarily, since whatever small values we assign to them we can always so adjust dx as to satisfy (8*031), the condition to which the displacement has to conform. Hence we infer
F,-^i=0 . . . . (8-034)
oz
94
THEORETICAL PHYSICS
[Ch. V
In order therefore that the particle may be in equilibrium the impressed force, F = {F^, Fy, F^) must satisfy the equations (8*032) and (8*034), or, what amounts to the same thing,
F F F
(8*035)
d(f> d<f) dcf) • • ' ' dx dy dz
Consider next the case where a particle is constrained to keep to a curve. Suppose the latter to be the intersection of two surfaces,
^ (x, y, z) = 0,
rp{x,y,z)=:0 (8*04)
The virtual displacement, 61 = {dx, dy, dz) has consequently to satisfy the conditions
and we infer that
OX dy dz
OX dy dz
(8-041)
^'-'fz
(8*05)
or, what amounts to the same thing,
F ^ ^
^' dx dx
F ^ ^
^' dy' dy
F ^ ^
^' dz' dz
= 0 .
(8*051)
The principle of virtual displacements may be illustrated by the following examples : —
Let the particle be confined to a spherical surface, but other- wise perfectly free, and suppose the force impressed on it to be directed vertically downwards. It might, for instance, be its weight. Let the origin of co-ordinates be at the centre of the
§8-1] PRINCIPLES OF DYNAMICS 95
sphere and the X axis have the direction of the force. Equation (8*031) becomes
xdx + ydy + zdz = 0, and for (8-032) and (8-034) we have
F^ = ?^x,
F, = h. Now since Fy = F, = 0,
F^ = 2.x, 0 = 2y, 0 = 2z, and as F^ is not zero, A cannot vanish, and therefore
y =z = 0. Consequently a; = + r or — r, where r is the radius of the sphere, or the possible positions of equilibrium are the uppermost and lowermost points on the sphere.^
An instructive example is that of a rigid body which can turn freely about a fixed axis, which we shaU take to be the Z axis of rectangular co-ordinates. The conditions for equili- brium are expressed by equations (8-01), the forces, F, being the impressed forces : not those due to constraints, together with the equations describing the constraints. These latter are, for every particle, s,
Szs = 0,
where d<j) is the same for all the particles in the body. Equation (8-01) therefore becomes
6<l>-^(x^,, - yJPJ = 0.
Now d^ is arbitrary, hence
2(a:,i^» - yj«) = 0.
This means that the sum of the moments of all the impressed forces with respect to the Z axis is zero, a result we have already obtained by a different method (§ 6-1).
§ 8-1. Peinciple of d'Alembert
The principle of virtual displacements is a statical one. It provides a means of investigating the conditions necessary for
Fx Fv Fz
^ Alternatively, the equations (8*035) become — = — = — and as
Fy = Fz^ ^it follows that 2/ = 2: = 0.
96 THEORETICAL PHYSICS [Ch. V
the equilibrium of a dynamical system. Its scope can be extended, however, by a device due to d'Alembert (Traite de Dynamique, 1743), so as to furnish a wider principle which constitutes a basis for the general investigation of the behaviour of djmamical systems.
Let F = (F^, Fy, FJ be the resultant force exerted on a particle of mass m, then
rr£^ -F,=0, dt^ ^ '
d^Z -r,
We may express these equations in the single statement
{-§ - ^0 '' + ("S - ^0'^ + (™S - ^^^ = ' ^'-'^
if (dx, dy, dz) is an arbitrary small displacement, since this necessi- tates the vanishing of the coefficients of dx, by and bz. Now (8*1) can be extended to apply to a system of particles, subject possibly to constraints, in the following way :
If the system should be subject to constraints, F^ will signify the force impressed on the particle, s, and will not include the force or forces due to the constraints, and the virtual displace- ments {bx'^, bys, bZg) are not all arbitrary, but subject to the equations defining the constraints.
Equation (8*1 1), with the interpretation just given, expresses the principle of d'Alembert.
A simple illustration of the principle is furnished by the
example of the rigid body in § 8. The procedure here differs
d A only in the substitution of ^^W-^ — -^sa? for the F^^ of § 8, and
(tt
corresponding expressions for F^y and F^^. We thus get
§8-2] PHINCIPLES OF DYNAMICS 97
and therefore, on account of the arbitrariness of dcj),
same thing as
which is the same thing as
di
(see § 6).
§ 8-2. Generalized Co-ordinates
We shall now introduce the generalized co-ordinates of § 6. The rectangular co-ordinates of any particle, s, of a system may be expressed in terms of the generalized co-ordinates, q, in the form
^s =fs tei, ^2, . . -qj,
Vs = ds (^1. ^2, . . . gj, ^s = K (s'l. 9^2, .. • gj,
in which /g, g^ and ^^ are given functions of the g's and the inherent constants of the dynamical system. We have in consequence
dx,=^dq,-^ ^dq,^- , . . ^^^J^dq^, dqi dq^ dq^
and similar equations for dy^ and dz^. It is convenient to use the symbol Xg itself to represent the functional dependence of the co-ordinate, x^, on the g's. We therefore obtain
dx, = ^dqi + ^dq^ + . . . + ^%„, ^qi ^^2 ^qn
dVs = -^dq, + -^dq^ + . . . + ^dq^, agi dq^ dq^
CZ VZ cz
dz,=^dq.+^dq, + ...+^dq^. . (8-2)
The symbol ' d ' will be used for increments which occur during the actual motion of the system, or during any motion we may tentatively ascribe to the system in the process of dis- covering the character of the actual motion. The symbol ' 6 ' will be used for virtual displacements and the increments depend- ing on them. The components, dxi, dyi, dzi; dx^, dyz, dz^\ ... of the virtual displacements of the particles of the system are not in general all arbitrary, as they may be subject to certain constraints. On the other hand the components, dq-i_, dq^, . . . dq^^ of a virtual displacement of the system are obviously quite
98 THEORETICAL PHYSICS [Ch. V
arbitrary, since the generalized co-ordinates are in fact so chosen as to be independent of one another. They thus satisfy the conditions imposed by the constraints, as it were, automatically.
If we replace the (i's in (8*2) by ^'s we get a corresponding set of equations for the virtual displacement {dx^, dy^, dz^) of a particle in terms of the associated dq's.
The velocity of the particle, s, is given, in terms of the gener- alized velocities, by
where q means -^. There are similar equations for -|^ and dt dt
dz
~. In these equations it will be observed that each differential dt
cX cl/ quotient, — , ^, . . . is expressed as a function of the ^'s and
constants inherent in the system (§ 5-3).
(jIT uij dz By squaring -— ?, -^-, -— respectively, adding and multiplying cit at u/t
by nis, the mass of the particle, we obtain twice its kinetic energy.
Therefore if T represents the kinetic energy of the system
2T = Qiiqiqi + 0i2gi^2 + . . . + Qmidn
+ —
+ Q^lMl + • - + QnnMn • (8'22)
in which ©12, for example, means
Q,, = E^ip p + lyi p + ^N . (8-221) ' (^qi^q^ oq^dq^ oqioq^)
Each Q is therefore expressed as a function of the g's and the inherent constants and it will be noticed that Q^^ = Q^^.
It is convenient to abbreviate (8*22) by writing it in the form
2T = Q^M^ .... (8-222)
in which the summation is indicated by the duplication or repe- tition of each of the subscripts a and p, and not by the symbol SS.
a ^
We see that (8-22) can be written in the form
2^ =^igi +^92^2 + . . -Mn • • • (8-23) or briefly . 2T = pjq^
where p, = Q,4^ + ^^2^2 + . • • + Qan% • • (8*24)
or i>, = Qa^q^'
§8-2] PRINCIPLES OF DYNAMICS 99
The quantity p^ is the generalized momentuin correspond- ing to the co-ordinate q^. Differentiating 2T partially with respect to g„, we get (see § 6-4),
dT
^« = ai: («-2«)
dT If, for instance, we take ^— - , we might carry out the differentia-
cq2
tion firstly along the second row of (8 '22), thus obtaining and then along the second vertical column, obtaining
612^1 + ^22^2 + Q^^iz + . . . Qn4n'
The two expressions are equal to one another (since Q^^ = Q^^) and together make 2jp^. Therefore
in agreement with (8*241).
It is important to note that (8*22) expresses 2T as a function of the q's, q's and the inherent constants of the system. It is a quadratic function of the g's. In (8*24) each p is also expressed as a function of the g's, g's and the inherent constants. It is a linear function of the g's.
From (8*24) we derive the equations
qi = I^llPl + ^12P2 + . . . + BlnPn, q^ = B^^p^ + R22P2 + . . . + R2nVn,
in = KiPl + ^n2P^ + . . . + B^nPn ' • (8*25)
in which the R^^ sue functions of the g's and inherent constants, and B^p — B^^. If we use the symbol i Q\ for the determinant
Qllj ^125 • • • Qin
^2l5 ^2 2? • • • ^2n
and the symbol \Q\a^ for the determinant which is formed by omitting the row, a, and the column, ^, each of these sub-deter- minants or minors having its sign so adjusted that, for example
101 =QM\.i + QM\..+ . . . +QM\^n (8-251)
then JJ„^=1^ (8-252)
100 THEORETICAL PHYSICS [Ch. V
Substituting the expressions (8*25) for q^ in (8*23), we get
+ E21P2P1 + R22P2P2 + . . . -\-R2np2pn
+
+ KlPnPl + I^n^PnP^ + • • • + RnnPnPn (8*26)
Which expresses 2T as a function of the g^'s, the ^'s and the inherent constants.
From equation (8*26) we get
dT
^- = E^^pi + K2P2 + • ' ' + KnPn
GPa
'ir'' ^'-''^
by a process similar to that used to derive (8*241 ).
The work done, during a small displacement of a system, is
and therefore, when we substitute for the dx^, dy^, dz^ the expres- sions in (8*2), we get
<t>idqi + (/»2^g2 + . . . + (l>ndqn, or ct>Jq^ (8-28)
in which it is easily seen that
We may term ^1, (1)2, etc. the generalized forces corresponding to the co-ordinates q^, q^, etc.
6 =I^{f ^ 4- F ^' -\- F ~'\ (8-281)
§ 8-3. Principle op Ekeegy
The use of the term ' energy ' is of comparatively recent origin ; but the conception of energy began to emerge as far back as the time of Huygens (1629-1695). In § 5-1 we deduced from Newton's laws that the increase of the kinetic energy, ^mv^, of a iDarticle is equal to the work done by, the force or forces acting on it (5*12). In any mechanical system whatever the work done by the forces of the system is equal to the corre- sponding increase of its kinetic energy. This is the principle of energy in the form (principle of vis viva) which is of peculiar importance in mechanics, i.e. in connexion with problems in which we are concerned only with movements of material masses under the influence of forces, explicitly given, or due to con- straints or analogous causes.
§8-3] PRINCIPLES OF DYNAMICS 101
The further development of the conception of energy is linked up with discoveries in different directions. In many mechanical problems, some of which we have already dealt with, the work done by the forces can be equated to the decrement of a certain quantity, V (§ 5-1), a function of the co-ordinates of the system, or a function, we may say, of its configuration. Since this is equal to the increment of the kinetic energy, T, a function of the state of motion of the system and its configuration, the sum T -\- V remains unaltered. The work is done at the expense of V and results in an equal increase of T. Then the consistent failure of all attempts to devise a machine (perpetuum mobile) capable of doing work gratis, and the success, on the other hand, in devising machines capable of doing work by the consumption of coal, gas or oil, gradually produced the conviction that work can be done only at some expense ; that whenever work is done, something is necessarily consumed. This something is called energy, and we conventionally adopt the amount of work done as a measure of the energy consumed. This does not mean that the energy of a body or a system is merely its capacity for doing work. There is some reason to believe that energy has a more substantial character, more perseity than is suggested by ' capacity for doing work '.
Finally the experimental work of a long line of investigators, Count Rumford, Davy, Colding, Hirn and above all. Joule, established that when heat ^ is generated by doing work, as for example in overcoming friction, and alternatively when work is done, as in the case of the steam engine, at the expense of heat, the quantity of heat (generated or consumed as the case may be ) is proportional to the work done ; the factor of proportionality (mechanical equivalent of heat) being the same, whether work is done at the expense of heat or heat produced in consequence of work done. This suggested that the heat in a body should be identified with the kinetic energy (or kinetic and potential energy) of the particles (molecules) of which it is constituted, and gave rise to the modern Principle of Conservation of Energy, according to which the energy in the world remains invariable in quantity. The constancy of ^ + F in certain mechanical systems is merely a special case therefore of the wider energy principle, and in the middle period of last century, and still later, it was generally held that, not only heat, but all other forms of energy were either kinetic energy or potential energy in the sense in which these terms are used in mechanics.
The principle of conservation of energy is in excellent accord with the view, which until quite recent times was universally
^ Heat measured by the use of mercury thermometers. See § 15" 5.
102 THEORETICAL PHYSICS [Ch. V
held, that physical and chemical phenomena are au fond mechanical phenomena ; and almost till the closing years of the century physical theories were held to be satisfactory or other- wise, just in proportion to the degree of success with which they furnished a mechanical picture of the Newtonian type.
§ 8.4. Equations o^ Hamilton and Lagrange
If a function
F = F{?., 2., • • • ffJ ■ • • • (8-4) exist, such that
where </>!, ^2? • • • ^^ are the generalized forces (8*28), the work done by them, during a small displacement of the system, will be
^y^ ^y 1 ^y 7 ^oA^^^
- ^li - -^k^ ... - ^qn . 8-402
dV or - ^qa>
This must be equal to the corresponding increase of the kinetic energy, T, Therefore
dT = -^^dq^ (8-41)
and in consequence of (8*4)
dT = -dV or d(T -]-V) =0;
so that the mechanical energy, T -\- V, remains constant. Such a system is said to be conservative. This is the exceptional case. In general T -\- V varies. This may happen in conse- quence of a complementary variation of the T -{- V oi some other system, or it may be associated with the development of heat, as when there are frictional forces, or with variations of other forms of energy.
Instead of confining our attention to conservative systems, let us suppose that there is a potential energy function, F, such that the generalized forces are given by (8*401) ; but that V has the form
y = F(2i, q,, . . . q, t) ... (8-42)
|
-=tv |
+ '> |
|
d{T + V) - |
-'>■ |
|
d(T + V) _ dt |
dv dt ■ |
§8-4] PRINCIPLES OF DYNAMICS i03
Equation (8'41) will stiU hold, but since
we have
-^-w^ = Tt • • • • («-42i)
As the time does not appear explicitly in the expression for T, whether we take (8-22) or (8-26), we get from (8-421)
dt dt
When T is expressed as a function of the generalized mo- menta (8*26 ) we shall represent J' + F by the symbol H, so that H does not merely denote the energy, T -{- V, but it is also a functional symbol. Since V does not contain the p's it is clear that
(8-423) cp^ cp^
and therefore by (8-27)
|
dH _dT dp, dp, dH '^ - dp. |
|
|
dq, dH dt dp. |
or .^ _ ^ (8-43)
It is essential that we should bear in mind that the partial
dT differential quotient — , T being expressed as a fu7iction of the
dT p's and q^s, is quite different from - — obtained from T expressed
as a function of the q's and q's. In fact, the former differentiation is subjected to the condition that the ^'s and the remainder of the g''s are kept constant, while the latter is subjected to the condition that the g's and the remainder of the g's are kept constant. To avoid confusion let us write
dT{p, q) ^DT dq ~ Dq
and dTii^^dT
dq dq
104 THEORETICAL PHYSICS [Ch. V
We may express a small change dT in the kinetic energy of a system in the following different ways : —
2dT = pMa + q.^Pa .... (by 8-23)
dT dT
^^=f>+f>- • • • («-^^^
DT
Subtracting the last of these from the first, and replacing-—-
by q, (8-27), we get
dT==pMa--^dq^, . . . .(8-441)
and the second equation (8*44) may be expressed in the form
dT=pMa + ^^dq^ . . . (8-442)
dT since Pa=;^—' Hence by comparison of (8*441) and (8*442) dq^
we find
f: = -^ (8-45)
Adding ^r— to both sides of this equation, we obtain
D[T + F) _ _ d[T - V) Dqa. dq,
which we may put in the form
^^=-f (8-451)
dqa ^a
where X = J' — F is also a functional symbol indicating that T is expressed as a function of the g's and g's. On the left of this equation D/Dq^ has been replaced by d/dq^, since the functional symbol H akeady indicates that the T in it is a function of the g's and ^'s.
From (8-41) we have
dt dq^ dt therefore
dT dp^ D{T + V) dq, _
dp, dt Dq, dt
§8-5]
PRINCIPLES OF DYNAMICS
105
and replacing ^— by ■— (8-27), we obtain
\dt ~^ dqj dt '
and consequently also
/dp^ _^_L\dq^ ^
\ dt dqj dt (See 8-451). This suggests, though it does not prove, the equations
dt dq^
and
dt dq^
(8-46)
(8-461)
Their validity will be established in § 8-6. The equations 8-46 together with (8-43) are known as Hamilton's canonical equations. The equations (8-461) are the equations of Lagrange and are usually written in the form
dt\dqj dq^
(8-462)
We may write them in this way, because V does not contain the g's and therefore
dT _ d{T -V) _dL
Ma Ma Ma
The function L is called the Lagrangian Function.
Pc
§ 8-5. Illustrations. Cyclic Co-ordikates
As a first illustration we may take the case of the compound pendulum § 6-5. Here we have one q, which is, conveniently, the angle 0, Fig. 6-5.
The energy equation is (6-501)
i/C^y - Mgh cos 6> = ^ . . . (8-5) and system is conservative. The corresponding p is
^=W='dt • • • • (8-501)
106 THEORETICAL PHYSICS [Ch. V
Therefore H = ^- Mgh cos d
and the canonical equations are
| = -|(g_M..c,..),
-7- = ?r-( — ^ — -^^Q^ COS 0 ) ; dt dp\2I ^ ) '
whence we obtain
dS ^p
It~I and therefore
/^ = - Mgh sin e,
in agreement with (6*5 ).
The Lagrangian function for the pendulum is
L = im + Mgh cos d,
and consequently the Lagrangian equation is
d^d whence /— - + Mgh sin 0 = 0
dt^
as before.
It will be noticed that when there are n degrees of freedom there are 2n canonical equations of the first order while there are n Lagrangian equations of the second order.
The case of the pendulum is merely illustrative. It is clear that nothing is gained by the equations of Hamilton or Lagrange in cases like this. Having set up the energy equation, it can only be described as a retrograde step to differentiate it. It is when we come to systems with more than one degree of freedom that the merits of the methods of Hamilton and Lagrange begin to appear.
Let us turn to the case, § 5-5, of a particle moving under
the influence of a central force — . We get (see 5*51) for the Hamiltonian function
H=f + /^+? .... (8-51)
one of the q's is the radial distance, r, and the other is the angle, d.
37 ^^ {¥^^ + ^^^ cos d) —^ {1/02 + Mgh cos 0} = 0,
§8-5] PRINCIPLES OF DYNAMICS 107
In this case Hamilton's equations become
dt dr\2m 2mr^ r J'
dpe _ _i/^_^ JV__^^\ dt dd\2m 2mr^ r )'
dt dp\2m 2mr^ r )'
dt djpe\2m 2mr^ r )' On carrying out the partial differentiations we get
dp, _ Po^ B dt mr^ r^'
dr _ p^ dt m ' dd _ Pe dt mr^'
From these four first order equations we may derive the following two second order equations :
d^r pe^ , B dt^ mr^ r^
^\mr^^\ =0 (8-52)
dt\ dt) ^ ^
This example illustrates two points : (i) The two equations we have obtained are sufficient, since the object may be said to be to express r and d as functions of the time. We have already one equation, the energy equation, at the very outset, and therefore we do not need both the equations (8*52) which we have derived. Instead of employing for the final solution of the dynamical problem the equations (8*52), it is preferable to use the energy equation and one of them. The reason for this is that the energy equation has already advanced one step in the series of integrations marking the way to the final goal, the accompanying constant of integration being in fact the most important of all, namely the energy, (ii) Whenever one or more of the co-ordinates do not appear explicitly in the function H, as for example 6 in the problem of the motion of a particle under a central force, the corresponding momentum is constant. Such co-ordinates are termed cyclic co-ordinates.
|
108 THEORETICAL PHYSICS |
|
|
The Lagrangian function derived from (5-51) is |
|
|
B |
|
|
■D or L = Imr^ + imr^S^ - -. |
|
|
Therefore |
|
|
dL |
|
|
'^=mr^e, |
|
|
^ = -^^+^' |
|
|
dd "' |
|
|
and consequently dV ., , B |
|
|
i (m.^5) = 0, |
[Ch. V
in agreement with (8'52).
For another illustration we may turn to the problem of the spinning top. The energy equation (see § 7-3) gives us
+ iC {^ + cos dip}^ + Mgh cos d = E . . (8-53) from which we find
p^ = ^^sin^ d.y) -{- C{^ + cos d.y)) cos 6, Pe = ^0,
P4> = (^{^ + cos d.y)}. Hence
and
i: = i^(sin2 (9.^2 ^ 192) ^ ic'(^ _!_ ^os 6).y»)2 - iff^^ cos d (8-532)
Whether we employ the equations of Hamilton or those of Lagrange, we find
p^ = constant, p^ = constant,
and these, together with the energy equation (8*53) are equiv- alent to (7-31), (7*32) and (7'33) which we have found already. The preceding examples illustrate conservative systems, in
§8-6] PRINCIPLES OF DYNAMICS 109
which the potential energy, V, does not contain the time. The following example furnishes a simple illustration of a non- conservative Hamiltonian system. A particle of mass m is constrained to keep to a straight line, and subject to a restoring force proportional to its displacement from a fixed point, 0, plus a force which is a simple harmonic function of the time. Its equation of motion wiU be
m— -| = — /Ltq + R C0& cot
where ^, R and co are constants. In this case
and
^ ^ 2m V = ~q^ — qB cos cot
since F is defined to fulfil the condition The Hamiltonian function is therefore
force = —
2m
^g2 — qR cos oyi,
and the Lagrangian function,
\mq^ — ^q^ + qR cos oyt.
§ 8-6. Principles of Action
If we have to deal with a system of not more than two degrees of freedom we may repre- sent its configuration and be- haviour graphically, by rectan- gular axes of co-ordinates using lengths measured from the origin along two of the axes to repre- sent the values of the g's and a length measured along the re- maining axis to represent the corresponding time (Fig. 8-6). The motion of the system will be completely represented by a
line such as (1, 2) in the diagram. We shall use the methods and the language which are appropriate for this graphical repre- sentation for systems of any number of degrees of freedom.
The principle of d'Alembert (8-11), if applied to the type
Fig. 8-6
110 THEORETICAL PHYSICS [Ch. V
of djmamical system dealt with in §§8-4 and 8-5, will take the form
In this equation the summations are sufficiently indicated by the repetitions of s and a. The s summation extends over all the particles of the system. The dx^, dy^, . . . dq^ . . . repre- sent virtual displacements. Our purpose is to investigate the actual motion of the system (represented by the path (1, 2) in Fig. 8-6) by studying its relation to motions represented by neighbouring paths (such as that shown in the figure by a broken line). These lines are comparable with the d lines of Fig. 3-31 in the proof of the theorem of Stokes, and we may conveniently suppose them to be drawn on a surface. It is helpful to regard the virtual displacements, dq^, as given by the intersections of this surface by a family of surfaces,
f(qi,q., . . .q,,t)^c . . . . (8-61)
These are quite arbitrarily chosen surfaces on account of the
arbitrariness of the virtual displacements, dq^. If we pick out
one of them by giving the constant C any value A, a neighbouring
surface will be one for which
C = K+dL
The lines of intersection of this family of surfaces with the
surface on which the d lines lie we shall naturally call 6 lines,
as in § 3-3. The symbol 6 will therefore represent an increment
incurred in passing along a 5 line from the d line representing
the actual motion to the neighbouring d line ; while the symbol
d will represent an increment incurred in passing along a d line
from a surface C = A to a neighbouring surface C = A + (iA,
i.e. from one 5 line to the next.
We shall now make use of a device, already employed in
previous investigations, namely that embodied in the formula
db diah) . da
a—- = \ ^ — 6 -— .
dt dt dt
Substituting for a and 6 respectively,
OX, and m,— ^, 'dt
we get
and similar expressions for
(J IJ (J z
in.-~^dy, and m,——^dz,. ' dt^ ^' 'dt^ '
§8-6] PRINCIPLES OF DYNAMICS
Equation (8*6 ) thus becomes
Ill
dt\ 'dt ' ^ 'dt dV ''
'dt ^' 'dt '
dy.d ,^ , , dz.d,^ , 'dt dr ^'^ dt dV '^
0,
or, (§ 8-2),
dV dql
dq^ = 0 (8-612)
We have seen (3'314) that ddx = ddx, etc., and we may show in a similar way that ddt = ddt ; but it does not follow, for d /dx\
^mple, that -r{^^) = ^(t")' ^® ^^^^ ^^ ^^^^ (^i§- ^*^1)
exai
or 61
/dx\ _ (^(a; + (5:r) \di) ~
/(Za;\ _ \^/ ~ /dx\ _ (i(5a; W / ~ ~di
dx dt'
d{t + dt) dt ddx — dx ddt dt dt + dt ddt ddx dx ddt dt~df and similar formulae for 'dy\ , . /dz^
(8-62)
<i) "" <i)^
jr+^j:+d/a:+Jjt:j
pc^doo
Fig. 8-61
With the aid of (8*62) we may now express (8*612) in the form
(XiX „
\^M.)
dt
m„
and therefore (§ 8-4)
\dt J 'dt \dtj 'dt \dtj
/dx,\^ , /dy\^ , /dz\^\ddt , 37.
0,
<^w!'-
ST -2T'^ + dV -^-^c dt dt
and by (8-421
l^ipM.}-ST-2Tf + SV-§St
= 0,
0.
112
Now
THEORETICAL PHYSICS
dr ^ dt dt '
[Ch. V
therefore we find
'^^ipM.-m-ST-2T'^ + SV-,E'^
0,
or
{Pa^qa
Mt
Edt} - d(2T -E) - (2T - E)~ = 0.
U/t
If we multiply this equation by dt and integrate between the limits 1 and 2 we obtain
2 2
Edt
VMa
or finally
2 pMa - Edt
- [ {dtd{2T -E) -\- {2T - E)5dt} = 0,
d[{2T -E)dt = 0 . . (8-63)
In this equation the variations symbolized by 6 are subject to no conditions, except that they are smaU.
Let us give our attention in the first place to conservative systems, i.e. systems for which dE = 0. Since the variations in (8-63) are arbitrary we may subject them to the condition SE = db constant. We then have
2 2 2 2
or
pMa - Edt
PaK
-I
2Tdt
SE^dt -{-E 1
ddt = 0,
+ SEit^ -t^) =6 \2Tdt
(8-631)
If we suppose the two paths to join at the lower limit 1 but not at the upper limit 2, we get, on dropping the index 2,
2Tdt. . . (8-632)
Padq. + (t- t,)dE = d^
or, if we use the symbol A for the integral on the right,
Pjqa + (i- ii)dE = dA ... (8-633) The function A is one of those to which the term action is applied and (8-633) indicates that it may be expressed as a function of the g's and E, and therefore
dA
Pa =.
t. =
dq:
M
dE
(8-634)
§8-6] PRINCIPLES OF DYNAMICS 113
If the system is strictly periodic and the range of integration,
2
, extends exactly over the period, r, of the system, 1
2
the terms | Pa^q^ I must vanish, and we find
(^2 - t^)dE = SA, or if we denote this particular value of A by the letter J,
If :'y (8-^35)
In the next place let us suppose the two paths to be co-
2
terminous in space (not necessarily in time) so that | Pa^^a I = 0,
1 since the terminal dq's vanish. Then if the variations are subjected to the condition dE = 0 we find
{2Tdt = 0 (8-636)
1
for systems for which dE = 0, i.e. for conservative systems. This is the principle of least action in its original form. It was first given in 1747 by de Maupertuis, a Frenchman, who was, for a time, president of the Royal Prussian Academy during the reign of Frederick the Great. He claimed for his principle a foundation in the attributes of the Deity. ' Notre principe ... est une suite necessaire de I'emploi le plus sage de cette puissance,' i.e. ' la puissance du Createur,' and the principle has turned out to be not unworthy of the claim made for it. A better name for it would be ' principle of stationary action ',
2
since the action \2Tdt is not in aU cases a minimum.
\
1 If in (8*63) we suppose the two paths to be co-terminous in space and time, i.e. the terminal variations dq^ and dt are all zero, we get
2
d [{2T -E)dt = 0
1
2
or
[ (T - V)dt = 0 .... (8-64)
114 THEORETICAL PHYSICS [Ch. V
This form (the most important one) of the principle of action is known as Hamilton's principle. The function
S =
{ (T - V)dt (8-65)
is called Hamilton's principal function, while the function A (8*633) is called Hamilton's characteristic function.
If we take the two paths to be co-terminous (in space and time) at the lower limit only we get from (8*63), dropping the upper index, 2,
Pa^qa -ESt = dS (8-66)
and therefore >S is a function of the g's and the time and
^-|'-^=S • • • («-^")
We may use Hamilton's principle (8*64) to establish the canonical equations and the equations of Lagrange. If we express E as a function of the ^'s, q's and t it becomes
2
or